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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. [From T. D. Noe, Nov 04 2009]

LINKS

A. Weimholt, Table of n, a(n) for n = 1..10000

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-Fran├žois 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: A080639 A186306 A047483 * 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

Corrected information about the tau(n)+3 orderly numbers -- T. D. Noe, Nov 16 2009

STATUS

approved

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Last modified August 17 13:03 EDT 2017. Contains 290635 sequences.