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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
 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, 59, 61, 67, 68, 71, 72, 73, 76, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 118, 124, 127, 131, 133, 137, 139, 149, 151, 157, 158, 162, 163, 164, 167, 173, 177, 178, 179 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS n: {divisors(n)} == {1,2,...,tau(n)} mod k   -------------------------------------------    1: {1} == {1} mod 2    2: {1,2} == {1,2} mod 3    5: {1,5} == {1,2} mod 3    7: {1,7} == {1,2} mod 5    8: {1,2,8,4} == {1,2,3,4} mod 5    9: {1,9,3} == {1,2,3} mod 7   11: {1,11} == {1,2} mod 3 or 9   12: {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7   13: {1,13} == {1,2} mod 11   17: {1,17} == {1,2} mod 3,5, or 15   19: {1,19} == 1,2 mod 17   20: {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7   23: {1,23} == {1,2} mod 3,7, or 21   27: {1,27,3,9} == {1,2,3,4} mod 5   29: {1,29} == {1,2} mod 3,9, or 27   31: {1,31} == {1,2} mod 29   37: {1,37} == 1,2 mod 5,7, or 35   38: {1,2,38,19} == {1,2,3,4} mod 5   41: {1,41} == {1,2} mod 3,13, or 39   43: {1,43} == {1,2} mod 41   47: {1,47} == {1,2} mod 3,5,9,15, or 45   52: {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7   53: {1,53} == {1,2} mod 3,17, or 51   57: {1,57,3,19} == {1,2,3,4} mod 5   58: {1,2,58,29} == {1,2,3,4} mod 5   59: {1,59} == {1,2} mod 3,19, or 57   61: {1,61} == {1,2} mod 59   67: {1,67} == {1,2} mod 5,13, or 65   68: {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7   71: {1,71} == {1,2} mod 3,23, or 69   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   73: {1,73} == {1,2} mod 71   76: {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7   79: {1,79} == {1,2} mod 7,11, or 77   83: {1,83} == {1,2} mod 3,9,27, or 81   87: {1,87,3,29} == {1,2,3,4} mod 5   89: {1,89} == {1,2} mod 3,29, or 87   97: {1,97} == {1,2} mod 5,19, or 95 The primes other than 3 are orderly. Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7. For primes, k values can be p-2 or a divisor of p-2 other than 1. 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. The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7. 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). 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. 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 LINKS A. Weimholt, Table of n, a(n) for n = 1..10000 Bill McEachen, A167408/A002858 EXAMPLE 12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and    1 == 1 (mod 7)    2 == 2 (mod 7)    3 == 3 (mod 7)    4 == 4 (mod 7)   12 == 5 (mod 7)    6 == 6 (mod 7) MATHEMATICA 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 *) CROSSREFS Cf. A167409 = very orderly numbers (k = tau(n) + 1). Cf. A167410 = disorderly numbers = numbers not in this sequence. Cf. A167411 = minimal k values for the orderly numbers. Sequence in context: A186306 A047483 A339309 * A047388 A284529 A191767 Adjacent sequences:  A167405 A167406 A167407 * A167409 A167410 A167411 KEYWORD nonn,nice AUTHOR Andrew Weimholt, Nov 03 2009 EXTENSIONS Minor editing by N. J. A. Sloane, Nov 06 2009 Information about the tau(n)+3 orderly numbers corrected by T. D. Noe, Nov 16 2009 STATUS approved

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Last modified May 13 08:08 EDT 2021. Contains 343836 sequences. (Running on oeis4.)