[seqfan] The Wichita Algorithm.
Robert Wilson
rgwv at rgwv.com
Mon Oct 30 21:06:26 CET 2023
Dear Michael,
I believe that we have missed a few sequences which involve the Wichita algorithm.
Edit as you please. I'm still calculating.
I would like to get A000001 submitted if only partially tomorrow.
Sincerely yours, Bob.
%I A000001
%N A000001 Number of distinct primes p utilized in the mapping of k = k - k/p terminating at 1; where p is any of the prime factors of k.
%S A000001 0, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 2, 3, 4, 3, 4, 1, 4, 2, 4, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 4, 5, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 4, 5, 3, 4, 4, 3, 1, 4, 4, 5, 2, 5, 4, 5, 2, 3, 3, 3, 3, 5, 3, 4, 2, 2, 3, 4, 3, 3, 4, 4, 3, 4, 3, 4, 4, 4, 5, 4, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4
%O A000001 1,3
%C A000001 Obviously A001221(n) <= a(n) <= pi(n) = A000720(n).
%C A000001 Just because some prime < n is not a prime factor of n does not preclude it from being a prime utilized in the cascade from n to 1. Take for instance n=14; 14 -> 12 and 3 is a prime factor of 12 but not of 14.
%C A000001 If p is a prime factor of n, then a(p^e * n) = a[n), where e is any exponent.
%C A000001 The number of primes p counting multiplicity is obviously the same as the path length, A332810. For n>1, 2 is always one of the primes.
%C A000001 Records: 1, 2, 3, 7, 23, 47, 139, 283, 659, 1319, 8637, 20147, 83491, 169861, 999013, ..., .
%D A000001
%H A000001
%f A000001
%e A000001 a(1) = 0 because 1 is at the end of all iterations;
%e A000001 a(2) = 1 since 2 -> 2 - 2/2 = 1, so there is one iteration of the mapping and it only involves the prime 2;
%e A000001 a(3) = 2 since 3 -> 2 -> 1 and this involves two primes, 2 and 3;
%e A000001 a(7) = 3 since 7 -> 6 -> 3 or 2 and this involves three primes, 7, 3, and 2; etc.
%t A000001 a[n_] := Block[{m = n, p, lst = {}}, While[m > 1, p = FactorInteger[m][[1, 1]]; AppendTo[lst, p]; m = m - m/p]; Length@ Union@ lst]; Array[a, 105]
%Y A000001 Cf. Records A000002
%K A000001
%A A000001 Michael De Vlieger and RGWv, 21 Oct 2023.
%E A000001
I was hoping to see some application here, but no! This calculation >= A333123
f[n_] := Block[{m = n, p, lst = {}}, While[m > 1, p = FactorInteger[m][[1, 1]]; AppendTo[lst, p]; m = m - m/p]; Apply[Multinomial, Length@# & /@ Split at Sort@ Most@ lst]];
f = {1, 1, 1, 1, 2, 2, 6, 1, 3, 3, 12, 3, 12, 12, 12, 1, 4, 6, 30, 4, 30, 20, 120, 4, 10, 20, 10, 20, 120, 20, 120, 1, 120, 5, 120, 10, 60, 60, 60, 5, 30, 60, 420, 30, 60, 210, 1680, 5, 210, 15, 30, 30, 210, 20, 105, 30, 140, 210, 1680, 30, 210, 210, 140, 1, 210, 210, 1680, 6, 1680, 210}
A333123 = {1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 5, 5, 1, 1, 5, 5, 3, 10, 5, 5, 4, 3, 7, 5, 9, 9, 12, 12, 1, 17, 2, 21, 9, 9, 14, 16, 4, 4, 28, 28, 9, 21, 14, 14, 5, 28, 7, 7, 12, 12, 14, 16, 14, 28, 23, 23, 21, 21, 33, 42, 1, 33, 47, 47, 3, 61, 56}
As you can see, there is no relationship between the two sequences.
%I A000002
%N A000002 Records of A000001
%S A000002 1, 2, 3, 7, 23, 47, 139, 283, 659, 1319, 8637, 20147, 83491, 169861, 999013, 3797401, 8153567, 55443959, 187814447, 646611887
%O A000002 0,2
%C A000002 Also the first time occurrence of n in A000001.
%D A000002
%H A000002
%f A000002
%e A000002
%t A000002 f[n_] := Block[{m = n, p, lst = {}}, While[m > 1, p = FactorInteger[m][[1, 1]]; AppendTo[lst, p]; m = m - m/p]; Length@ Union@ lst]; t[_] := 0; k = 1; lst = {}; While[k < 10000001, a = f at k; If[t[a] == 0, t[a] = k; AppendTo[lst, k]]; k++]; lst
%Y A000002
%K A000002
%A A000002
%E A000002
%I A000003
%N A000003 The sum of primes p not counting multiplicity utilized in the mapping of k = k - k/p terminating at 1; where p is any of the prime factors of k.
%S A000003 0, 2, 5, 2, 7, 5, 12, 2, 5, 7, 18, 5, 18, 12, 10, 2, 19, 5, 24, 7, 12, 18, 41, 5, 7, 18, 5, 12, 41, 10, 41, 2, 21, 19, 17, 5, 42, 24, 18, 7, 48, 12, 55, 18, 10, 41, 88, 5, 12, 7, 22, 18, 71, 5, 18, 12, 24, 41, 100, 10, 71, 41, 12, 2, 23, 21, 88, 19, 44, 17, 88
%O A000003 1,2
%C A000003
%D A000003
%H A000003
%f A000003
%e A000003
%t A000003 a[n_] := Block[{m = n, p, lst = {}}, While[m > 1, p = FactorInteger[m][[1, 1]]; AppendTo[lst, p]; m = m - m/p]; Total@ Union@ lst]; Array[a, 71]
%Y A000003
%K A000003
%A A000003
%E A000003
%I A000004
%N A000004
%S A000004 0, 2, 5, 4, 9, 7, 14, 6, 10, 11, 22, 9, 22, 16, 14, 8, 25, 12, 31, 13, 19, 24, 47, 11, 18, 24, 15, 18, 47, 16, 47, 10, 27, 27, 23, 14, 51, 33, 27, 15, 56, 21, 64, 26, 19, 49, 96, 13, 28, 20, 30, 26, 79, 17, 31, 20, 36, 49, 108, 18, 79, 49, 24, 12, 31, 29, 96, 29, 52, 25
%O A000004 1,2
%C A000004 Every integer value shows up except for 1 and 3.
%C A000004 Except for one, all records are at a prime index.
%C A000004 Number of record prime indices < 10^k: 4, 14, 30, 61, 119, 224, 337, 512, 814, 1287, ..., .
%D A000004
%H A000004
%f A000004
%e A000004
%t A000004 a[n_] := Block[{m = n, p, t = 0}, While[m > 1, p = FactorInteger[m][[1, 1]]; t += p; m = m - m/p]; t]; Array[a, 70]
%Y A000004
%K A000004
%A A000004
%E A000004
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3987113279, 4019210879, 4079476799, 4119102719, 4135742399, 4157399999, 4204561919, 4258264799, 4273846079, 4336385759, 4369291679, 4370446079, 4397955839, 4434271679, 4505873279, 4523567039, 4527423359, 4529622719, 4566931679, 4606330559, 4643909759, 4683735359, 4698635519, 4778581439, 4780059359, 4783971359, 4785164159, 4799836799, 4802938079, 4809383039, 4809875039, 4815635999, 4818523679, 4820682719, 4824827999, 4864868159, 4869037919, 4873620959, 4879038239, 4879945919, 4901721599, 4913938559, 4925238719, 4974061439, 4979895359, 4988961599, 5041508159, 5045654399, 5089451999, 5098644479, 5106916799, 5118030719, 5152667519, 5157114239, 5202307679, 5206259519, 5265095519, 5316268319, 5317388159, 5325044639, 5334352319, 5340086399, 5347224959, 5369108159, 5421625439, 5425989599, 5474891519, 5485821119, 5495345279, 5545151999, 5557444799, 5636100959, 5643139199, 5719161599, 5758286399, 5816781119, 5873981759, 5900770559, 5936823359, 5989521599, 5996971199, 6025621439, 6061350719, 6077673599, 6085273919, 6093575039, 6096193919, 6105909119, 6133196159, 6254915039, 6258453599, 6275121599, 6328583519, 6330292799, 6351346559, 6364565759, 6386244479, 6423133439, 6437208959, 6492003839, 6514728959, 6571712639, 6644713919, 6665070719, 6688822079, 6709214399, 6726984959, 6820139519, 6854889599, 6860383679, 6945219839, 6954712319, 6979304639, 6987447359, 7018198079, 7070345279, 7077683519, 7107810239, 7130181119, 7171882559, 7204729919, 7243716479, 7302173279, 7352142719, 7352649119, 7364107679, 7366772159, 7421494079, 7533633599, 7556989439, 7611914879, 7639423679, 7652263679, 7687991999, 7724777279, 7760376959, 7931808959, 7944409919, 7945905599, 7978983359, 8019718079, 8032347839, 8051351039, 8087217599, 8179017599, 8260840319, 8278602239, 8282453759, 8358113279, 8380133759, 8447663999, 8481686399, 8491095359, 8502626879, 8554752959, 8558522879, 8631543359, 8772193439, 8776035839, 8779954559, 8781722399, 8790856799, 8795511359, 8805419999, 8806719359, 8820246719, 8825478239, 8827274399, 8832211679, 8834094719, 8841006719, 8935862399, 8942114879, 9036256319, 9069193919, 9142779839, 9178498559, 9225914879, 9272504639, 9290627519, 9295453439, 9310932479, 9437167679, 9457706879, 9499303679, 9619750079, 9631271999, 9641365439, 9683593919, 9697728959, 9738075839, 9766162559, 9815519999, 9821011199, 9847944959, 9948122879, 10034656319, 10045226879, 10067318399, 10090694399, 10137304319, 10187002559, 10197195839, 10236061439, 10384210559, 10404615359, 10493358719, 10519262399, 10589495999, 10598756159}
%I A000005
%N A000005 Records of A333123.
%S A000005 1, 6, 12, 14, 21, 30, 33, 35, 42, 62, 63, 66, 69, 77, 105, 129, 141, 154, 161, 209, 213, 215, 217, 231, 301, 322, 329, 434, 437, 462, 469, 473, 483, 645, 651, 658, 693, 805, 893, 966, 973, 987, 989, 1333, 1407, 1419, 1645, 1771, 1943, 1946, 1974, 1977, 1981, 2021, 2841, 2881, 2967
%O A000005 1,2
%C A000005 Only 18 terms are even.
%C A000005
%D A000005
%H A000005
%f A000005
%e A000005
%t A000005 a[n_] := Block[{m = n, p, t = 0}, While[m > 1, p = FactorInteger[m][[1, 1]]; t += p; m = m - m/p]; t]; Array[a, 70]
%Y A000005
%K A000005
%A A000005
%E A000005
{1, 6, 12, 14, 21, 30, 33, 35, 42, 62, 63, 66, 69, 77, 105, 129, 141, 154, 161, 209, 213, 215, 217, 231, 301, 322, 329, 434, 437, 462, 469, 473, 483, 645, 651, 658, 693, 805, 893, 966, 973, 987, 989, 1333, 1407, 1419, 1645, 1771, 1943, 1946, 1974, 1977, 1981, 2021, 2841, 2881, 2967, 3311, 3619, 4042, 4277, 4613, 4935, 4945, 4991, 5313, 5931, 5934, 5937, 5969, 5977, 6063, 6923, 8319, 9177, 9499, 10011, 10105, 10199, 10857, 12169, 13839, 13853, 14147, 18753, 18791, 19411, 20769, 21747, 21791, 22043, 22231, 26273, 27699, 28294, 28337, 34615, 36507, 37527, 37563, 38399, 42441, 45563, 46483, 56717, 58233, 62651, 66693, 70735, 76153, 85011, 85183, 91601, 93013, 94987, 99029, 101563, 106099, 116607, 116879, 119239, 133859, 135407, 139449, 143491, 155617, 198359, 212221, 212359, 234041, 265119, 268001, 268793, 272083, 279039, 279887, 309071, 311563, 311707, 325381, 406221, 418817, 422389, 433951, 466851, 537887, 538403, 560057, 571943, 618611, 623887, 624073, 637077, 651091, 651751, 815323, 834673, 883177, 934689, 937013, 947849, 976143, 1152093, 1177633, 1177867, 1249093, 1283387, 1301853, 1304161, 1304491, 1331839, 1671367, 1767247, 1767343, 1872219, 1876007, 1898579, 1955253, 1959209, 2265541, 2277667, 2357053, 2504019, 2568093, 2572733, 2574451, 2604223, 2610301, 2665699, 3300293, 3433679, 3483479, 3533601, 3611527, 3800039, 3912483, 3913473, 3938843, 4003601, 4562257, 5214007, 5317853, 5369649, 5382943, 5493313, 5927593, 6182239, 6257909, 6259841, 7071159, 7092263, 7169261, 7746487, 7830893, 7830903, 7883663, 7997097, 8245069, 9129127, 9131437, 10817119, 10972761, 10999927, 11873203, 12370729, 12371401, 12383269, 12529181, 13452481, 14117257, 14228053, 14320383, 14349401, 14650229, 15680747, 15812059, 16050911, 16513333, 18272107, 18659893, 22023343, 23650989, 23764423, 23774743, 24735207, 24755311, 24779059, 24785329, 25304941, 25603109, 28742329, 29074717, 29322689, 32531699, 33043363, 33744637, 37376503, 38414369, 38441441, 38453309, 43327273, 43667317, 44039611, 46274321, 47585563, 49498113, 49571647, 49595719, 50638219, 50648281, 55979679, 58707371, 60097597, 61311077, 69458903, 69472439, 69494153, 75266097, 75719833, 75754027, 76809327, 76964969, 78578501, 86720293, 87400897, 87968067, 89233213, 90695759, 94560569, 99797969, 101298583, 101353279, 111981589, 112242943, 112356377, 115593331, 122684147, 138991079, 139050299, 150580997, 151623007, 153804633, 157276241, 176560021, 178601833, 189264629, 205258823, 230765253, 230836497, 230894907, 231466549, 236212459, 253076429, 269090087, 269173163, 278354351, 281634199, 303895749, 303951679, 304059837, 304967309, 308700079, 352306219, 353493319, 358877477, 369170953, 410951597, 429177539, 465136117, 471828723, 472996861, 486459071, 535805499, 538754783, 557131091, 588871507} (610000000)
Even terms: {6, 12, 14, 30, 42, 62, 66, 154, 322, 434, 462, 658, 966, 1946, 1974, 4042, 5934, 28294}
AHP Venture, LLC
Robert G. Wilson V
1082 N.W. 139th Terrace
Pembroke Pines, FL 33028-2340
Cel: (316) 640-9841
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