a brief look at sum of prime factors
OK. A fun one today. Well, I think it is fun, a mathematician would call it trivial. Sum of prime factors. Way back I called these things strange-integers, and they served as an early test of my code. Simply enough define:
strange-int[x] = the sum of the prime factors of x.
Say r has prime factorisation:
r = p1^n1 * p2^n2 * p3^n3 * p4^n4 * ...
Then strange-int[r] = p1*n1 + p2*n2 + p3*n3 + p4*n4 + ...
It has these properties:
strange-int[p] = p if p is prime, or p = 4
strange-int[p] < p otherwise
This has the effect that we can form a tree of integers (for the impatient, here is an image of that tree), where each application of strange-int[] steps one step closer to a final prime.
Here are the first twenty:
-- we need this due to a quirk of my code.
-- where the strange-int on the right hand side is a function-operator in my functions code.
sa: strange-int |*> #=> strange-int |_self>
sa: table[number,strange-int] range(|number: 2>,|number: 20>)
+--------+-------------+
| number | strange-int |
+--------+-------------+
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 5 |
| 7 | 7 |
| 8 | 6 |
| 9 | 6 |
| 10 | 7 |
| 11 | 11 |
| 12 | 7 |
| 13 | 13 |
| 14 | 9 |
| 15 | 8 |
| 16 | 8 |
| 17 | 17 |
| 18 | 8 |
| 19 | 19 |
| 20 | 9 |
+--------+-------------+
Next we can take a look using the strange-int-list operator. This maps an integer to a list of integers down to the final prime. Here are the first 30:
sa: strange-int-list |*> #=> strange-int-list |_self>
sa: how-long-is-strange-int-list |*> #=> how-many strange-int-list |_self>
sa: table[number,strange-int,strange-int-list,how-long-is-strange-int-list] range(|number: 2>,|number: 30>)
+--------+-------------+------------------+------------------------------+
| number | strange-int | strange-int-list | how-long-is-strange-int-list |
+--------+-------------+------------------+------------------------------+
| 2 | 2 | 2 | 1 |
| 3 | 3 | 3 | 1 |
| 4 | 4 | 4 | 1 |
| 5 | 5 | 5 | 1 |
| 6 | 5 | 6, 5 | 2 |
| 7 | 7 | 7 | 1 |
| 8 | 6 | 8, 6, 5 | 3 |
| 9 | 6 | 9, 6, 5 | 3 |
| 10 | 7 | 10, 7 | 2 |
| 11 | 11 | 11 | 1 |
| 12 | 7 | 12, 7 | 2 |
| 13 | 13 | 13 | 1 |
| 14 | 9 | 14, 9, 6, 5 | 4 |
| 15 | 8 | 15, 8, 6, 5 | 4 |
| 16 | 8 | 16, 8, 6, 5 | 4 |
| 17 | 17 | 17 | 1 |
| 18 | 8 | 18, 8, 6, 5 | 4 |
| 19 | 19 | 19 | 1 |
| 20 | 9 | 20, 9, 6, 5 | 4 |
| 21 | 10 | 21, 10, 7 | 3 |
| 22 | 13 | 22, 13 | 2 |
| 23 | 23 | 23 | 1 |
| 24 | 9 | 24, 9, 6, 5 | 4 |
| 25 | 10 | 25, 10, 7 | 3 |
| 26 | 15 | 26, 15, 8, 6, 5 | 5 |
| 27 | 9 | 27, 9, 6, 5 | 4 |
| 28 | 11 | 28, 11 | 2 |
| 29 | 29 | 29 | 1 |
| 30 | 10 | 30, 10, 7 | 3 |
+--------+-------------+------------------+------------------------------+
Now, lets sort by strange-int-list length, and find the top 100 of those in the first 100,000 integers.
sa: strange-int |*> #=> strange-int |_self>
sa: strange-int-list |*> #=> strange-int-list |_self>
sa: length-of-strange-int-list |*> #=> how-many strange-int-list |_self>
sa: table[number,strange-int,strange-int-list,length-of-strange-int-list] select[1,100] reverse sort-by[length-of-strange-int-list] range(|number: 2>,|number: 100000>)
+--------+-------------+---------------------------------------------------------+----------------------------+
| number | strange-int | strange-int-list | length-of-strange-int-list |
+--------+-------------+---------------------------------------------------------+----------------------------+
| 55694 | 27849 | 55694, 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 27933 | 9314 | 27933, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 99895 | 19984 | 99895, 19984, 1257, 422, 213, 74, 39, 16, 8, 6, 5 | 11 |
| 97629 | 4659 | 97629, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 97533 | 10843 | 97533, 10843, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 92915 | 18588 | 92915, 18588, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 90177 | 30062 | 90177, 30062, 15033, 5014, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 86696 | 10843 | 86696, 10843, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 86662 | 43333 | 86662, 43333, 2566, 1285, 262, 133, 26, 15, 8, 6, 5 | 11 |
| 84934 | 42469 | 84934, 42469, 6074, 3039, 1016, 133, 26, 15, 8, 6, 5 | 11 |
| 83718 | 4659 | 83718, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 74924 | 18735 | 74924, 18735, 1257, 422, 213, 74, 39, 16, 8, 6, 5 | 11 |
| 74416 | 4659 | 74416, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 69765 | 4659 | 69765, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 68653 | 5294 | 68653, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 11 |
| 55686 | 9286 | 55686, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 11 |
| 46405 | 9286 | 46405, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 11 |
| 46142 | 23073 | 46142, 23073, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 11 |
| 36398 | 18201 | 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 11 |
| 30993 | 10334 | 30993, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 11 |
| 27849 | 9286 | 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 11 |
| 9314 | 4659 | 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 99853 | 7694 | 99853, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 10 |
| 99763 | 1556 | 99763, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 99482 | 49743 | 99482, 49743, 5533, 514, 259, 44, 15, 8, 6, 5 | 10 |
| 98422 | 49213 | 98422, 49213, 1726, 865, 178, 91, 20, 9, 6, 5 | 10 |
| 98386 | 49195 | 98386, 49195, 9844, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 98284 | 24575 | 98284, 24575, 993, 334, 169, 26, 15, 8, 6, 5 | 10 |
| 98214 | 16374 | 98214, 16374, 2734, 1369, 74, 39, 16, 8, 6, 5 | 10 |
| 98151 | 32720 | 98151, 32720, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 97618 | 48811 | 97618, 48811, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 97209 | 1556 | 97209, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 96523 | 13796 | 96523, 13796, 3453, 1154, 579, 196, 18, 8, 6, 5 | 10 |
| 96467 | 13788 | 96467, 13788, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 95962 | 47983 | 95962, 47983, 3704, 469, 74, 39, 16, 8, 6, 5 | 10 |
| 95562 | 5317 | 95562, 5317, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 94593 | 31534 | 94593, 31534, 15769, 1226, 615, 49, 14, 9, 6, 5 | 10 |
| 94469 | 5574 | 94469, 5574, 934, 469, 74, 39, 16, 8, 6, 5 | 10 |
| 94426 | 1556 | 94426, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 94222 | 47113 | 94222, 47113, 4294, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 93494 | 46749 | 93494, 46749, 15586, 7795, 1564, 44, 15, 8, 6, 5 | 10 |
| 93414 | 15574 | 93414, 15574, 614, 309, 106, 55, 16, 8, 6, 5 | 10 |
| 93382 | 46693 | 93382, 46693, 934, 469, 74, 39, 16, 8, 6, 5 | 10 |
| 92702 | 46353 | 92702, 46353, 15454, 7729, 190, 26, 15, 8, 6, 5 | 10 |
| 92654 | 46329 | 92654, 46329, 15446, 7725, 116, 33, 14, 9, 6, 5 | 10 |
| 92244 | 7694 | 92244, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 10 |
| 92156 | 23043 | 92156, 23043, 7684, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 91445 | 18294 | 91445, 18294, 3054, 514, 259, 44, 15, 8, 6, 5 | 10 |
| 91398 | 15238 | 91398, 15238, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 91285 | 18262 | 91285, 18262, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 91213 | 1774 | 91213, 1774, 889, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 91198 | 45601 | 91198, 45601, 1502, 753, 254, 129, 46, 25, 10, 7 | 10 |
| 89974 | 44989 | 89974, 44989, 6434, 3219, 69, 26, 15, 8, 6, 5 | 10 |
| 89702 | 44853 | 89702, 44853, 14954, 7479, 286, 26, 15, 8, 6, 5 | 10 |
| 89222 | 6382 | 89222, 6382, 3193, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 89038 | 44521 | 89038, 44521, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 88870 | 8894 | 88870, 8894, 4449, 1486, 745, 154, 20, 9, 6, 5 | 10 |
| 87985 | 17602 | 87985, 17602, 692, 177, 62, 33, 14, 9, 6, 5 | 10 |
| 87938 | 43971 | 87938, 43971, 14660, 742, 62, 33, 14, 9, 6, 5 | 10 |
| 87554 | 43779 | 87554, 43779, 14596, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 87222 | 14542 | 87222, 14542, 674, 339, 116, 33, 14, 9, 6, 5 | 10 |
| 86934 | 14494 | 86934, 14494, 7249, 670, 74, 39, 16, 8, 6, 5 | 10 |
| 86408 | 1556 | 86408, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 85983 | 28664 | 85983, 28664, 3589, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 85686 | 14286 | 85686, 14286, 2386, 1195, 244, 65, 18, 8, 6, 5 | 10 |
| 85497 | 28502 | 85497, 28502, 14253, 4754, 2379, 77, 18, 8, 6, 5 | 10 |
| 85317 | 28442 | 85317, 28442, 14223, 445, 94, 49, 14, 9, 6, 5 | 10 |
| 85299 | 28436 | 85299, 28436, 7113, 2374, 1189, 70, 14, 9, 6, 5 | 10 |
| 84944 | 5317 | 84944, 5317, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 84939 | 1257 | 84939, 1257, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 84684 | 7064 | 84684, 7064, 889, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 83686 | 41845 | 83686, 41845, 8374, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 83466 | 4645 | 83466, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 10 |
| 83374 | 41689 | 83374, 41689, 934, 469, 74, 39, 16, 8, 6, 5 | 10 |
| 83282 | 41643 | 83282, 41643, 674, 339, 116, 33, 14, 9, 6, 5 | 10 |
| 82556 | 20643 | 82556, 20643, 993, 334, 169, 26, 15, 8, 6, 5 | 10 |
| 81845 | 16374 | 81845, 16374, 2734, 1369, 74, 39, 16, 8, 6, 5 | 10 |
| 80381 | 11490 | 80381, 11490, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 80269 | 11474 | 80269, 11474, 5739, 1916, 483, 33, 14, 9, 6, 5 | 10 |
| 79635 | 5317 | 79635, 5317, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 79302 | 13222 | 79302, 13222, 614, 309, 106, 55, 16, 8, 6, 5 | 10 |
| 79146 | 4405 | 79146, 4405, 886, 445, 94, 49, 14, 9, 6, 5 | 10 |
| 78747 | 26252 | 78747, 26252, 6567, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 78609 | 26206 | 78609, 26206, 13105, 2626, 116, 33, 14, 9, 6, 5 | 10 |
| 78502 | 39253 | 78502, 39253, 2326, 1165, 238, 26, 15, 8, 6, 5 | 10 |
| 78362 | 39183 | 78362, 39183, 393, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 78038 | 39021 | 78038, 39021, 13010, 1308, 116, 33, 14, 9, 6, 5 | 10 |
| 77845 | 15574 | 77845, 15574, 614, 309, 106, 55, 16, 8, 6, 5 | 10 |
| 77709 | 25906 | 77709, 25906, 12955, 2596, 74, 39, 16, 8, 6, 5 | 10 |
| 77642 | 38823 | 77642, 38823, 12944, 817, 62, 33, 14, 9, 6, 5 | 10 |
| 77396 | 1774 | 77396, 1774, 889, 134, 69, 26, 15, 8, 6, 5 | 10 |
| 77391 | 8605 | 77391, 8605, 1726, 865, 178, 91, 20, 9, 6, 5 | 10 |
| 77289 | 25766 | 77289, 25766, 1006, 505, 106, 55, 16, 8, 6, 5 | 10 |
| 77109 | 25706 | 77109, 25706, 12855, 865, 178, 91, 20, 9, 6, 5 | 10 |
| 76870 | 7694 | 76870, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 10 |
| 76363 | 10916 | 76363, 10916, 2733, 914, 459, 26, 15, 8, 6, 5 | 10 |
| 76338 | 4249 | 76338, 4249, 614, 309, 106, 55, 16, 8, 6, 5 | 10 |
| 76165 | 15238 | 76165, 15238, 422, 213, 74, 39, 16, 8, 6, 5 | 10 |
| 75722 | 37863 | 75722, 37863, 614, 309, 106, 55, 16, 8, 6, 5 | 10 |
| 74192 | 4645 | 74192, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 10 |
+--------+-------------+---------------------------------------------------------+----------------------------+
So it seems, in the first 100,000 integers, 55,694 and 27,933 are in some sense "the least prime", taking 12 steps of strange-int to reach a destination prime. And 91,198 is the only one in the top 100 that has 7 as the final prime instead of 5.
Now, let's upscale to the first 1,000,000 integers (I tried 10,000,000 but python seg-faulted at 1.4 million).
sa: strange-int-list |*> #=> strange-int-list |_self>
sa: length-of-strange-int-list |*> #=> how-many strange-int-list |_self>
sa: table[number,strange-int-list,length-of-strange-int-list] select[1,100] reverse sort-by[length-of-strange-int-list] range(|number: 2>,|number: 1000000>)
+--------+-------------------------------------------------------------------------+----------------------------+
| number | strange-int-list | length-of-strange-int-list |
+--------+-------------------------------------------------------------------------+----------------------------+
| 334142 | 334142, 167073, 55694, 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 14 |
| 921327 | 921327, 27933, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 13 |
| 868022 | 868022, 434013, 144674, 72339, 24116, 6033, 2014, 74, 39, 16, 8, 6, 5 | 13 |
| 823502 | 823502, 411753, 137254, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 13 |
| 723853 | 723853, 55694, 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 13 |
| 167073 | 167073, 55694, 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 13 |
| 999593 | 999593, 142806, 23806, 11905, 2386, 1195, 244, 65, 18, 8, 6, 5 | 12 |
| 991456 | 991456, 30993, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 982503 | 982503, 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 959566 | 959566, 479785, 95962, 47983, 3704, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 945958 | 945958, 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 944206 | 944206, 472105, 94426, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 929490 | 929490, 30993, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 922660 | 922660, 46142, 23073, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 12 |
| 912805 | 912805, 182566, 91285, 18262, 422, 213, 74, 39, 16, 8, 6, 5 | 12 |
| 904603 | 904603, 129236, 32313, 10774, 5389, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 896734 | 896734, 448369, 15490, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 892578 | 892578, 148768, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 892569 | 892569, 297526, 148765, 29758, 14881, 670, 74, 39, 16, 8, 6, 5 | 12 |
| 888542 | 888542, 444273, 148094, 74049, 24686, 12345, 831, 280, 18, 8, 6, 5 | 12 |
| 887613 | 887613, 295874, 147939, 4497, 1502, 753, 254, 129, 46, 25, 10, 7 | 12 |
| 882202 | 882202, 441103, 33944, 4249, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 873336 | 873336, 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 866890 | 866890, 86696, 10843, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 864627 | 864627, 288212, 72057, 24022, 12013, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 837277 | 837277, 119618, 59811, 19940, 1006, 505, 106, 55, 16, 8, 6, 5 | 12 |
| 780437 | 780437, 111498, 18588, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 774575 | 774575, 30993, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 745717 | 745717, 106538, 53271, 1982, 993, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 743815 | 743815, 148768, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 735006 | 735006, 122506, 61255, 12256, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 727780 | 727780, 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 723749 | 723749, 55686, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 719214 | 719214, 119874, 19984, 1257, 422, 213, 74, 39, 16, 8, 6, 5 | 12 |
| 697062 | 697062, 116182, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 688958 | 688958, 344481, 114830, 11490, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 674593 | 674593, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 668958 | 668958, 111498, 18588, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 650643 | 650643, 30993, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 645862 | 645862, 46142, 23073, 7694, 3849, 1286, 645, 51, 20, 9, 6, 5 | 12 |
| 638782 | 638782, 319393, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 638422 | 638422, 319213, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 631924 | 631924, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 613628 | 613628, 153411, 51140, 2566, 1285, 262, 133, 26, 15, 8, 6, 5 | 12 |
| 612505 | 612505, 122506, 61255, 12256, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 609218 | 609218, 304611, 101540, 5086, 2545, 514, 259, 44, 15, 8, 6, 5 | 12 |
| 606823 | 606823, 86696, 10843, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 601113 | 601113, 200374, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 599345 | 599345, 119874, 19984, 1257, 422, 213, 74, 39, 16, 8, 6, 5 | 12 |
| 592684 | 592684, 148175, 5937, 1982, 993, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 587427 | 587427, 195812, 48957, 16322, 8163, 913, 94, 49, 14, 9, 6, 5 | 12 |
| 580885 | 580885, 116182, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 575422 | 575422, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 568983 | 568983, 189664, 5937, 1982, 993, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 568196 | 568196, 142053, 47354, 23679, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 557465 | 557465, 111498, 18588, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 538414 | 538414, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 533198 | 533198, 266601, 88870, 8894, 4449, 1486, 745, 154, 20, 9, 6, 5 | 12 |
| 509446 | 509446, 36398, 18201, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 496966 | 496966, 248485, 49702, 24853, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 471633 | 471633, 157214, 78609, 26206, 13105, 2626, 116, 33, 14, 9, 6, 5 | 12 |
| 458187 | 458187, 152732, 38187, 4249, 614, 309, 106, 55, 16, 8, 6, 5 | 12 |
| 446466 | 446466, 74416, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 434013 | 434013, 144674, 72339, 24116, 6033, 2014, 74, 39, 16, 8, 6, 5 | 12 |
| 424113 | 424113, 141374, 70689, 23566, 11785, 2362, 1183, 33, 14, 9, 6, 5 | 12 |
| 417591 | 417591, 46405, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 411753 | 411753, 137254, 5294, 2649, 886, 445, 94, 49, 14, 9, 6, 5 | 12 |
| 401614 | 401614, 200809, 28694, 14349, 4786, 2395, 484, 26, 15, 8, 6, 5 | 12 |
| 390914 | 390914, 195459, 5937, 1982, 993, 334, 169, 26, 15, 8, 6, 5 | 12 |
| 372055 | 372055, 74416, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 371192 | 371192, 46405, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 367509 | 367509, 122506, 61255, 12256, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 360692 | 360692, 90177, 30062, 15033, 5014, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 353134 | 353134, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 343249 | 343249, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 334086 | 334086, 55686, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 306818 | 306818, 153411, 51140, 2566, 1285, 262, 133, 26, 15, 8, 6, 5 | 12 |
| 287773 | 287773, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 279044 | 279044, 69765, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 278405 | 278405, 55686, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 268342 | 268342, 134173, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 12 |
| 260079 | 260079, 86696, 10843, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 223239 | 223239, 74416, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 139526 | 139526, 69765, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 55694 | 55694, 27849, 9286, 4645, 934, 469, 74, 39, 16, 8, 6, 5 | 12 |
| 27933 | 27933, 9314, 4659, 1556, 393, 134, 69, 26, 15, 8, 6, 5 | 12 |
| 999242 | 999242, 499623, 166544, 1502, 753, 254, 129, 46, 25, 10, 7 | 11 |
| 998454 | 998454, 166414, 83209, 11894, 334, 169, 26, 15, 8, 6, 5 | 11 |
| 995216 | 995216, 62209, 8894, 4449, 1486, 745, 154, 20, 9, 6, 5 | 11 |
| 994966 | 994966, 71078, 5086, 2545, 514, 259, 44, 15, 8, 6, 5 | 11 |
| 994822 | 994822, 497413, 71066, 35535, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 994094 | 994094, 497049, 23679, 886, 445, 94, 49, 14, 9, 6, 5 | 11 |
| 993838 | 993838, 496921, 6070, 614, 309, 106, 55, 16, 8, 6, 5 | 11 |
| 993189 | 993189, 331066, 165535, 33112, 4145, 834, 144, 14, 9, 6, 5 | 11 |
| 992778 | 992778, 165468, 13796, 3453, 1154, 579, 196, 18, 8, 6, 5 | 11 |
| 992202 | 992202, 165372, 13788, 393, 134, 69, 26, 15, 8, 6, 5 | 11 |
| 990816 | 990816, 10334, 5169, 1726, 865, 178, 91, 20, 9, 6, 5 | 11 |
| 990588 | 990588, 82556, 20643, 993, 334, 169, 26, 15, 8, 6, 5 | 11 |
| 989593 | 989593, 89974, 44989, 6434, 3219, 69, 26, 15, 8, 6, 5 | 11 |
| 989513 | 989513, 141366, 23566, 11785, 2362, 1183, 33, 14, 9, 6, 5 | 11 |
+--------+-------------------------------------------------------------------------+----------------------------+
I guess one observation to make here is that length-of-strange-int-list grows rather slowly. And that 334,142 is in some sense the least prime.
Plenty more way's to slice and dice strange-ints. eg, count how many integers map to a given prime. Here is a simple example, making use of strange-int-prime that maps numbers to their primes:
-- show strange-int-prime in use:
sa: strange-int-prime |number: 55694>
|number: 5>
sa: strange-int-prime |number: 91198>
|number: 7>
-- now count the strange-int-primes:
sa: table[number,coeff] strange-int-prime range(|number: 2>,|number: 20>)
+--------+-------+
| number | coeff |
+--------+-------+
| 2 | 1 |
| 3 | 1 |
| 4 | 1 |
| 5 | 9 |
| 7 | 3 |
| 11 | 1 |
| 13 | 1 |
| 17 | 1 |
| 19 | 1 |
+--------+-------+
Now upscale to 100,000 and sort:
sa: table[number,coeff] select[1,100] coeff-sort strange-int-prime range(|number: 2>,|number: 100000>)
+--------+-------+
| number | coeff |
+--------+-------+
| 5 | 27023 |
| 7 | 15753 |
| 13 | 6821 |
| 11 | 6032 |
| 19 | 3837 |
| 17 | 3146 |
| 23 | 2289 |
| 31 | 1497 |
| 29 | 1194 |
| 43 | 1161 |
| 37 | 937 |
| 47 | 792 |
| 41 | 747 |
| 61 | 719 |
| 73 | 658 |
| 53 | 626 |
| 59 | 477 |
| 103 | 463 |
| 109 | 445 |
| 67 | 444 |
| 71 | 444 |
| 83 | 405 |
| 79 | 401 |
| 113 | 354 |
| 89 | 348 |
| 139 | 326 |
| 101 | 325 |
| 107 | 316 |
| 151 | 275 |
| 131 | 247 |
| 97 | 246 |
| 181 | 226 |
| 127 | 218 |
| 137 | 208 |
| 167 | 207 |
| 199 | 207 |
| 149 | 206 |
| 193 | 202 |
| 229 | 190 |
| 173 | 185 |
| 163 | 177 |
| 157 | 175 |
| 197 | 174 |
| 241 | 164 |
| 179 | 157 |
| 191 | 143 |
| 283 | 129 |
| 239 | 127 |
| 271 | 126 |
| 233 | 124 |
| 211 | 121 |
| 257 | 119 |
| 227 | 117 |
| 313 | 116 |
| 281 | 115 |
| 293 | 99 |
| 251 | 97 |
| 263 | 97 |
| 277 | 95 |
| 269 | 94 |
| 317 | 92 |
| 223 | 90 |
| 349 | 86 |
| 383 | 70 |
| 311 | 67 |
| 463 | 66 |
| 307 | 61 |
| 373 | 61 |
| 367 | 60 |
| 421 | 60 |
| 353 | 59 |
| 359 | 58 |
| 401 | 58 |
| 433 | 58 |
| 331 | 56 |
| 523 | 56 |
| 347 | 55 |
| 619 | 55 |
| 397 | 54 |
| 443 | 54 |
| 337 | 53 |
| 389 | 51 |
| 379 | 50 |
| 449 | 50 |
| 467 | 48 |
| 503 | 48 |
| 409 | 46 |
| 457 | 44 |
| 571 | 44 |
| 661 | 44 |
| 431 | 43 |
| 461 | 43 |
| 509 | 42 |
| 601 | 42 |
| 439 | 41 |
| 479 | 41 |
| 617 | 38 |
| 613 | 37 |
| 419 | 35 |
| 491 | 35 |
+--------+-------+
Now, upscale to 500,000:
sa: table[number,coeff] select[1,100] coeff-sort strange-int-prime range(|number: 2>,|number: 500000>)
+--------+--------+
| number | coeff |
+--------+--------+
| 5 | 133689 |
| 7 | 76847 |
| 13 | 33690 |
| 11 | 29661 |
| 19 | 18500 |
| 17 | 15111 |
| 23 | 11463 |
| 31 | 7730 |
| 29 | 6187 |
| 43 | 5479 |
| 37 | 4444 |
| 47 | 3879 |
| 41 | 3849 |
| 61 | 3503 |
| 73 | 3263 |
| 53 | 3096 |
| 103 | 2308 |
| 67 | 2304 |
| 109 | 2105 |
| 59 | 2097 |
| 71 | 2071 |
| 83 | 1926 |
| 79 | 1925 |
| 113 | 1655 |
| 139 | 1608 |
| 89 | 1607 |
| 107 | 1457 |
| 101 | 1278 |
| 151 | 1155 |
| 97 | 1148 |
| 181 | 1045 |
| 131 | 1022 |
| 199 | 991 |
| 193 | 955 |
| 137 | 903 |
| 229 | 901 |
| 167 | 889 |
| 173 | 852 |
| 157 | 821 |
| 197 | 821 |
| 127 | 795 |
| 163 | 787 |
| 149 | 784 |
| 179 | 771 |
| 241 | 748 |
| 233 | 708 |
| 283 | 686 |
| 191 | 660 |
| 271 | 652 |
| 313 | 650 |
| 281 | 616 |
| 227 | 615 |
| 239 | 591 |
| 263 | 569 |
| 211 | 559 |
| 257 | 558 |
| 251 | 538 |
| 277 | 530 |
| 317 | 516 |
| 269 | 499 |
| 223 | 484 |
| 349 | 467 |
| 293 | 459 |
| 311 | 450 |
| 421 | 412 |
| 463 | 412 |
| 353 | 399 |
| 383 | 397 |
| 359 | 386 |
| 433 | 381 |
| 401 | 375 |
| 307 | 374 |
| 389 | 366 |
| 337 | 359 |
| 373 | 350 |
| 467 | 347 |
| 379 | 346 |
| 619 | 345 |
| 347 | 339 |
| 397 | 330 |
| 331 | 327 |
| 449 | 320 |
| 523 | 316 |
| 367 | 315 |
| 443 | 302 |
| 509 | 298 |
| 461 | 293 |
| 661 | 290 |
| 439 | 288 |
| 419 | 284 |
| 409 | 283 |
| 571 | 272 |
| 601 | 268 |
| 479 | 264 |
| 491 | 263 |
| 431 | 262 |
| 503 | 257 |
| 457 | 256 |
| 643 | 255 |
| 617 | 245 |
+--------+--------+
And I guess we can ask the question, does this ordering of primes (5,7,13,11,19,17,23,31,29,43,37,...) stabilize as n -> infinity? I suspect it just might. OK. I had a closer look with more examples, and it might and it might not. If it does, it would provide a kind of natural ordering of primes. Whatever use that may be.
n from 2 to 100,000 gives this ordering:
5,7,13,11,19,17,23,31,29,43,37,47,41,61,73,53,59,103,109,67,71,83,79,113,89,139,101,107,151,131,97,...
n from 2 to 500,000 gives this ordering:
5,7,13,11,19,17,23,31,29,43,37,47,41,61,73,53,103,67,109,59,71,83,79,113,139,89,107,101,151,97,181,...
n from 500,000 to 1,000,000 gives this ordering:
5,7,13,11,19,17,23,31,29,43,37,41,47,61,73,53,103,71,109,67,59,83,79,113,139,89,107,101,151,97,181,...
n from 1,000,000 to 1,500,000 gives this ordering:
5,7,13,11,19,17,23,31,29,43,37,47,41,61,73,53,59,103,67,71,109,83,79,113,89,107,139,101,151,181,97,...
Next, we can use strange-int-prime to define an equivalency class:
x is in [p] if strange-int-prime[x] = p
Now, in BKO:
-- find strange-int-primes:
sa: strange-int-prime-op |*> #=> strange-int-prime |_self>
sa: map[strange-int-prime-op,strange-int-prime] range(|number: 2>,|number: 1000>)
-- find inverse-strange-int-primes:
sa: find-inverse[strange-int-prime]
-- find primes:
sa: is-prime-op |*> #=> is-prime |_self>
sa: map[is-prime-op,is-prime] range(|number: 2>,|number: 1000>)
-- find inverse-primes:
sa: find-inverse[is-prime]
-- for display reasons, just want the first 25 inverse-strange-int-primes:
sa: inverse-strange-int-prime-list |*> #=> select[1,25] inverse-strange-int-prime |_self>
-- now our pretty table:
sa: table[prime,inverse-strange-int-prime-list] select[1,75] inverse-is-prime |yes>
+-------+---------------------------------------------------------------------------------------------------------------------------+
| prime | inverse-strange-int-prime-list |
+-------+---------------------------------------------------------------------------------------------------------------------------+
| 2 | 2 |
| 3 | 3 |
| 5 | 5, 6, 8, 9, 14, 15, 16, 18, 20, 24, 26, 27, 33, 39, 44, 49, 51, 55, 62, 65, 66, 69, 70, 74, 77 |
| 7 | 7, 10, 12, 21, 25, 30, 32, 35, 36, 38, 42, 46, 50, 60, 64, 68, 72, 81, 87, 124, 129, 141, 152, 155, 158 |
| 11 | 11, 28, 40, 45, 48, 54, 86, 111, 115, 138, 164, 187, 215, 218, 226, 249, 258, 266, 287, 319, 329, 338, 380, 391, 407 |
| 13 | 13, 22, 56, 57, 63, 75, 80, 85, 90, 96, 102, 108, 121, 122, 146, 159, 166, 182, 212, 236, 260, 284, 308, 312, 314 |
| 17 | 17, 52, 88, 99, 147, 175, 194, 210, 224, 235, 250, 252, 282, 300, 320, 346, 360, 384, 405, 415, 432, 451, 466, 486, 498 |
| 19 | 19, 34, 93, 104, 117, 145, 165, 174, 176, 198, 245, 253, 289, 294, 303, 326, 350, 356, 420, 448, 452, 494, 500, 502, 504 |
| 23 | 23, 76, 136, 153, 219, 273, 302, 325, 355, 385, 390, 416, 426, 462, 468, 542, 550, 596, 655, 660, 686, 704, 706, 766, 786 |
| 29 | 29, 184, 207, 399, 475, 507, 543, 570, 595, 608, 684, 714, 715, 794, 847, 850, 858, 895 |
| 31 | 31, 58, 265, 318, 345, 368, 414, 517, 526, 561, 665, 697, 798, 833, 841, 845, 950 |
| 37 | 37, 248, 279, 435, 464, 522, 554, 759, 866, 867 |
| 41 | 41, 148, 651, 775, 930, 992 |
| 43 | 43, 82, 237, 296, 333, 662, 781, 879, 932, 957 |
| 47 | 47, 172, 328, 369, 734, 777, 835, 925 |
| 53 | 53, 376, 423, 842, 903 |
| 59 | 59, 424, 477 |
| 61 | 61, 118, 565, 678, 795, 848, 954 |
| 67 | 67, 488, 549, 885, 944 |
| 71 | 71, 268 |
| 73 | 73, 142, 417, 536, 603, 685, 822 |
| 79 | 79, 584, 657 |
| 83 | 83, 316, 939 |
| 89 | 89, 664, 747 |
| 97 | 97 |
| 101 | 101, 388 |
| 103 | 103, 202, 597, 776, 873, 985 |
| 107 | 107, 412, 808, 909 |
| 109 | 109, 214, 633, 824, 927 |
| 113 | 113, 436, 856, 963 |
| 127 | 127 |
| 131 | 131, 508 |
| 137 | 137 |
| 139 | 139, 274, 813 |
| 149 | 149 |
| 151 | 151, 298 |
| 157 | 157 |
| 163 | 163 |
| 167 | 167, 652 |
| 173 | 173 |
| 179 | 179 |
| 181 | 181, 358 |
| 191 | 191 |
| 193 | 193, 382 |
| 197 | 197, 772 |
| 199 | 199, 394 |
| 211 | 211 |
| 223 | 223 |
| 227 | 227, 892 |
| 229 | 229, 454 |
| 233 | 233, 916 |
| 239 | 239 |
| 241 | 241, 478 |
| 251 | 251 |
| 257 | 257 |
| 263 | 263 |
| 269 | 269 |
| 271 | 271, 538 |
| 277 | 277 |
| 281 | 281 |
| 283 | 283, 562 |
| 293 | 293 |
| 307 | 307 |
| 311 | 311 |
| 313 | 313, 622 |
| 317 | 317 |
| 331 | 331 |
| 337 | 337 |
| 347 | 347 |
| 349 | 349, 694 |
| 353 | 353 |
| 359 | 359 |
| 367 | 367 |
| 373 | 373 |
| 379 | 379 |
+-------+---------------------------------------------------------------------------------------------------------------------------+
The interpretation being all elements in the second column are members of the strange-int-prime equivalency class of the prime in the first column.
Finally, let's visualize our sum-of-prime factor tree:
sa: context sum of prime factors
sa: SoPF-op |*> #=> extract-value strange-int merge-labels(|number: > + |_self>)
sa: map[SoPF-op,SoPF] range(|2>,|1000>)
sa: save sum-of-prime-factors.sw
sa: q
$ ./sw2dot-v2.py sw-examples/sum-of-prime-factors.sw
Then open graph-examples/sum-of-prime-factors.dot with graphviz, producing:
NB: the picture is too large to see here, so click on it to see the full, but pretty, tree.
Anyway, the point of this post is not just about strange-ints. It is more about giving some examples of how you can slice and dice knowledge using the knowledge engine.
Home
previous: update on visualizing sw files
next: the easy way to make a big binary tree
updated: 19/12/2016
by Garry Morrison
email: garry -at- semantic-db.org
