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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. 7

%I

%S 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 59,61,67,68,71,72,73,76,79,83,87,89,97,101,103,107,109,113,117,118,

%U 124,127,131,133,137,139,149,151,157,158,162,163,164,167,173,177,178,179

%N 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 n: {divisors(n)} == {1,2,...,tau(n)} mod k

%C -------------------------------------------

%C 1: {1} == {1} mod 2

%C 2: {1,2} == {1,2} mod 3

%C 5: {1,5} == {1,2} mod 3

%C 7: {1,7} == {1,2} mod 5

%C 8: {1,2,8,4} == {1,2,3,4} mod 5

%C 9: {1,9,3} == {1,2,3} mod 7

%C 11: {1,11} == {1,2} mod 3 or 9

%C 12: {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7

%C 13: {1,13} == {1,2} mod 11

%C 17: {1,17} == {1,2} mod 3,5, or 15

%C 19: {1,19} == 1,2 mod 17

%C 20: {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7

%C 23: {1,23} == {1,2} mod 3,7, or 21

%C 27: {1,27,3,9} == {1,2,3,4} mod 5

%C 29: {1,29} == {1,2} mod 3,9, or 27

%C 31: {1,31} == {1,2} mod 29

%C 37: {1,37} == 1,2 mod 5,7, or 35

%C 38: {1,2,38,19} == {1,2,3,4} mod 5

%C 41: {1,41} == {1,2} mod 3,13, or 39

%C 43: {1,43} == {1,2} mod 41

%C 47: {1,47} == {1,2} mod 3,5,9,15, or 45

%C 52: {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7

%C 53: {1,53} == {1,2} mod 3,17, or 51

%C 57: {1,57,3,19} == {1,2,3,4} mod 5

%C 58: {1,2,58,29} == {1,2,3,4} mod 5

%C 59: {1,59} == {1,2} mod 3,19, or 57

%C 61: {1,61} == {1,2} mod 59

%C 67: {1,67} == {1,2} mod 5,13, or 65

%C 68: {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7

%C 71: {1,71} == {1,2} mod 3,23, or 69

%C 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 73: {1,73} == {1,2} mod 71

%C 76: {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7

%C 79: {1,79} == {1,2} mod 7,11, or 77

%C 83: {1,83} == {1,2} mod 3,9,27, or 81

%C 87: {1,87,3,29} == {1,2,3,4} mod 5

%C 89: {1,89} == {1,2} mod 3,29, or 87

%C 97: {1,97} == {1,2} mod 5,19, or 95

%C The primes other than 3 are orderly.

%C Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.

%C For primes, k values can be p-2 or a divisor of p-2 other than 1.

%C 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 The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.

%C 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 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 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 A. Weimholt, <a href="/A167408/b167408.txt">Table of n, a(n) for n = 1..10000</a>

%H Bill McEachen, <a href="/A167408/a167408.jpg">A167408/A002858</a>

%e 12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and

%e 1 == 1 (mod 7)

%e 2 == 2 (mod 7)

%e 3 == 3 (mod 7)

%e 4 == 4 (mod 7)

%e 12 == 5 (mod 7)

%e 6 == 6 (mod 7)

%t 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-Fran├žois Alcover_, Aug 19 2013 *)

%Y Cf. A167409 = very orderly numbers (k = tau(n) + 1).

%Y Cf. A167410 = disorderly numbers = numbers not in this sequence.

%Y Cf. A167411 = minimal k values for the orderly numbers.

%K nonn,nice

%O 1,2

%A _Andrew Weimholt_, Nov 03 2009

%E Minor editing by _N. J. A. Sloane_, Nov 06 2009

%E Information about the tau(n)+3 orderly numbers corrected by _T. D. Noe_, Nov 16 2009

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Last modified April 20 11:52 EDT 2021. Contains 343135 sequences. (Running on oeis4.)