A167411 -id:A167411 - cp's OEIS frontend

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.

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

Offset: 1

Andrew Weimholt, Nov 03 2009

   n: {divisors(n)} == {1,2,...,tau(n)} mod k
  -------------------------------------------
{1} == {1} mod 2
{1,2} == {1,2} mod 3
{1,5} == {1,2} mod 3
{1,7} == {1,2} mod 5
{1,2,8,4} == {1,2,3,4} mod 5
{1,9,3} == {1,2,3} mod 7
{1,11} == {1,2} mod 3 or 9
{1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
{1,13} == {1,2} mod 11
{1,17} == {1,2} mod 3,5, or 15
{1,19} == 1,2 mod 17
{1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
{1,23} == {1,2} mod 3,7, or 21
{1,27,3,9} == {1,2,3,4} mod 5
{1,29} == {1,2} mod 3,9, or 27
{1,31} == {1,2} mod 29
{1,37} == 1,2 mod 5,7, or 35
{1,2,38,19} == {1,2,3,4} mod 5
{1,41} == {1,2} mod 3,13, or 39
{1,43} == {1,2} mod 41
{1,47} == {1,2} mod 3,5,9,15, or 45
{1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
{1,53} == {1,2} mod 3,17, or 51
{1,57,3,19} == {1,2,3,4} mod 5
{1,2,58,29} == {1,2,3,4} mod 5
{1,59} == {1,2} mod 3,19, or 57
{1,61} == {1,2} mod 59
{1,67} == {1,2} mod 5,13, or 65
{1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
{1,71} == {1,2} mod 3,23, or 69
{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
{1,73} == {1,2} mod 71
{1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
{1,79} == {1,2} mod 7,11, or 77
{1,83} == {1,2} mod 3,9,27, or 81
{1,87,3,29} == {1,2,3,4} mod 5
{1,89} == {1,2} mod 3,29, or 87
{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

			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)

A. Weimholt, Table of n, a(n) for n = 1..10000
Bill McEachen, A167408/A002858

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.

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 *)

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

A167409 Very orderly numbers: a number N is "very orderly" if the set of the divisors of N is congruent to the set {1,2,...,tau(N)} mod (tau(N) + 1).

Original entry on oeis.org

1, 2, 5, 8, 11, 12, 17, 20, 23, 27, 29, 38, 41, 47, 52, 53, 57, 58, 59, 68, 71, 72, 76, 83, 87, 89, 101, 107, 113, 117, 118, 124, 131, 133, 137, 149, 158, 162, 164, 167, 173, 177, 178, 179, 188, 191, 197, 203, 218, 227, 233, 236, 237, 239, 243, 244, 247, 251, 257

Offset: 1

Andrew Weimholt, Nov 03 2009

nonn

The very orderly numbers are orderly numbers (cf. A167408) with K = tau(N) + 1.

Equivalently, all divisors must be pairwise distinct and distinct from 0, modulo tau(N) = number of divisors of N. - M. F. Hasler, Mar 21 2023

			12 is in the sequence as it has the 6 divisors {1, 2, 3, 4, 12, 6} which when reduced mod (6+1) give {1, 2, 3, 4, 5, 6} = {1, 2, ..., tau(12)}. - _David A. Corneth_, Mar 21 2023

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

Cf. A167408 (orderly numbers), A167410 (disorderly numbers).
Cf. A167411 (minimal K values for the orderly numbers).
Cf. A000005 (tau = number of divisors).

veryOrderlyQ[n_] := (If[tau = DivisorSigma[0, n]; Union[Mod[Divisors[n], tau + 1]] == Range[tau], Return[True]]; False); Select[ Range[260], veryOrderlyQ] (* Jean-François Alcover, Aug 19 2013 *)

select( {vo(n)=#(n=divisors(n))==#(n=Set(n%(1+#n))) && n[1]}, [1..999]) \\ M. F. Hasler; updated for current PARI syntax Mar 21 2023

A167410 Disorderly Numbers: numbers not in A167408 (orderly numbers).

Original entry on oeis.org

3, 4, 6, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 56, 60, 62, 63, 64, 65, 66, 69, 70, 74, 75, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108

Offset: 1

Andrew Weimholt, Nov 03 2009

nonn

			3 is disorderly because there exists no K > 2=tau(3), such that {1,3} == {1,2} mod K.

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

Cf. A167408 - Orderly Numbers
Cf. A167409 - Very Orderly Numbers ( K = tau(N)+1 )
Cf. A167411 - Minimal K Values for the Orderly Numbers

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[120], !orderlyQ[#]&] (* Jean-François Alcover, Nov 03 2016 *)

cp's OEIS Frontend

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.

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A167409 Very orderly numbers: a number N is "very orderly" if the set of the divisors of N is congruent to the set {1,2,...,tau(N)} mod (tau(N) + 1).

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A167410 Disorderly Numbers: numbers not in A167408 (orderly numbers).

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Mathematica