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A167408 Orderly numbers: a number n is orderly if there exists some number k > tau(n) such that the set of the divisors of n is congruent to the set {1,2,...,tau(n)} mod k.

%I A167408 #30 Mar 19 2023 20:20:03
%S A167408 1,2,5,7,8,9,11,12,13,17,19,20,23,27,29,31,37,38,41,43,47,52,53,57,58,
%T A167408 59,61,67,68,71,72,73,76,79,83,87,89,97,101,103,107,109,113,117,118,
%U A167408 124,127,131,133,137,139,149,151,157,158,162,163,164,167,173,177,178,179
%N A167408 Orderly numbers: a number n is orderly if there exists some number k > tau(n) such that the set of the divisors of n is congruent to the set {1,2,...,tau(n)} mod k.
%C A167408    n: {divisors(n)} == {1,2,...,tau(n)} mod k
%C A167408   -------------------------------------------
%C A167408    1: {1} == {1} mod 2
%C A167408    2: {1,2} == {1,2} mod 3
%C A167408    5: {1,5} == {1,2} mod 3
%C A167408    7: {1,7} == {1,2} mod 5
%C A167408    8: {1,2,8,4} == {1,2,3,4} mod 5
%C A167408    9: {1,9,3} == {1,2,3} mod 7
%C A167408   11: {1,11} == {1,2} mod 3 or 9
%C A167408   12: {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
%C A167408   13: {1,13} == {1,2} mod 11
%C A167408   17: {1,17} == {1,2} mod 3,5, or 15
%C A167408   19: {1,19} == 1,2 mod 17
%C A167408   20: {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
%C A167408   23: {1,23} == {1,2} mod 3,7, or 21
%C A167408   27: {1,27,3,9} == {1,2,3,4} mod 5
%C A167408   29: {1,29} == {1,2} mod 3,9, or 27
%C A167408   31: {1,31} == {1,2} mod 29
%C A167408   37: {1,37} == 1,2 mod 5,7, or 35
%C A167408   38: {1,2,38,19} == {1,2,3,4} mod 5
%C A167408   41: {1,41} == {1,2} mod 3,13, or 39
%C A167408   43: {1,43} == {1,2} mod 41
%C A167408   47: {1,47} == {1,2} mod 3,5,9,15, or 45
%C A167408   52: {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
%C A167408   53: {1,53} == {1,2} mod 3,17, or 51
%C A167408   57: {1,57,3,19} == {1,2,3,4} mod 5
%C A167408   58: {1,2,58,29} == {1,2,3,4} mod 5
%C A167408   59: {1,59} == {1,2} mod 3,19, or 57
%C A167408   61: {1,61} == {1,2} mod 59
%C A167408   67: {1,67} == {1,2} mod 5,13, or 65
%C A167408   68: {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
%C A167408   71: {1,71} == {1,2} mod 3,23, or 69
%C A167408   72: {1,2,3,4,18,6,72,8,9,36,24,12} == {1,2,3,4,5,6,7,8,9,10,11,12} mod 13
%C A167408   73: {1,73} == {1,2} mod 71
%C A167408   76: {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
%C A167408   79: {1,79} == {1,2} mod 7,11, or 77
%C A167408   83: {1,83} == {1,2} mod 3,9,27, or 81
%C A167408   87: {1,87,3,29} == {1,2,3,4} mod 5
%C A167408   89: {1,89} == {1,2} mod 3,29, or 87
%C A167408   97: {1,97} == {1,2} mod 5,19, or 95
%C A167408 The primes other than 3 are orderly.
%C A167408 Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.
%C A167408 For primes, k values can be p-2 or a divisor of p-2 other than 1.
%C A167408 T. D. Noe observed that for composite orderly numbers, n, k seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
%C A167408 The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.
%C A167408 The orderly numbers with k = tau(n)+3 come in many forms. See A168003. It appears that tau(n)+3 is a prime with primitive root 2 (A001122).
%C A167408 The forms for composite orderly numbers with k = tau(n)+1 are too numerous to list here, but seem to occur for any prime k > 3.
%C A167408 Let p be any prime. Then p^(m-2) is in this sequence if m is a prime with primitive root p. For example, 2^(m-2) is here for every m in A001122; 3^(m-2) is here for every m in A019334; 5^(m-2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m-2) here. All these numbers are actually very orderly (A167409) because we can choose k = tau(n)+1. - _T. D. Noe_, Nov 04 2009
%H A167408 A. Weimholt, <a href="/A167408/b167408.txt">Table of n, a(n) for n = 1..10000</a>
%H A167408 Bill McEachen, <a href="/A167408/a167408.jpg">A167408/A002858</a>
%e A167408 12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and
%e A167408    1 == 1 (mod 7)
%e A167408    2 == 2 (mod 7)
%e A167408    3 == 3 (mod 7)
%e A167408    4 == 4 (mod 7)
%e A167408   12 == 5 (mod 7)
%e A167408    6 == 6 (mod 7)
%t A167408 orderlyQ[n_] := (For[dd = Divisors[n]; tau = Length[dd]; k = 3, k <= Max[tau + 4, Last[dd] - 2], k++, If[ Union[ Mod[dd, k]] == Range[tau], Return[True]]]; False); Select[ Range[180], orderlyQ] (* _Jean-François Alcover_, Aug 19 2013 *)
%Y A167408 Cf. A167409 = very orderly numbers (k = tau(n) + 1).
%Y A167408 Cf. A167410 = disorderly numbers = numbers not in this sequence.
%Y A167408 Cf. A167411 = minimal k values for the orderly numbers.
%K A167408 nonn,nice
%O A167408 1,2
%A A167408 _Andrew Weimholt_, Nov 03 2009
%E A167408 Minor editing by _N. J. A. Sloane_, Nov 06 2009
%E A167408 Information about the tau(n)+3 orderly numbers corrected by _T. D. Noe_, Nov 16 2009