A167409 -id:A167409 - 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

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 *)

A167411 a(n) = the minimal K value for the orderly number A167408(n).

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 3, 7, 11, 3, 17, 7, 3, 5, 3, 29, 5, 5, 3, 41, 3, 7, 3, 5, 5, 3, 59, 5, 7, 3, 13, 71, 7, 7, 3, 5, 3, 5, 3, 101, 3, 107, 3, 7, 5, 7, 5, 3, 5, 3, 137, 3, 149, 5, 5, 11, 7, 7, 3, 3, 5, 5, 3, 179, 7, 3, 191, 3, 197, 5, 11, 5, 13, 3, 227, 3, 7, 5, 3, 239, 7, 7, 5, 3, 11, 3, 3, 5, 3

Offset: 1

Andrew Weimholt, Nov 03 2009

nonn

			a(6) = 7, because A167408(6) = 9, and divisors of 9 = {1,9,3} == {1,2,3} mod 7.

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. A167410 - Disorderly Numbers - numbers not in A167408.

orderlyQkValue[n_] := (For[dd = Divisors[n]; tau = Length[dd]; k = 2, k <= Max[tau + 4, Last[dd] - 2], k++, If[Union[Mod[dd, k]] == Range[tau], Return[{True, k}]]]; {False, 0}); A167411List = Select[{#, orderlyQkValue[#]}& /@ Range[400] , #[[2, 1]]&][[All, 2, 2]] (* Jean-François Alcover, Nov 03 2016 *)

A055720 Numbers k such that d(k)+1 | k.

Original entry on oeis.org

3, 4, 10, 15, 28, 35, 54, 55, 63, 64, 65, 85, 95, 98, 100, 115, 125, 135, 144, 145, 147, 155, 156, 175, 176, 185, 189, 205, 215, 234, 235, 245, 260, 265, 295, 297, 305, 335, 336, 351, 355, 364, 365, 395, 400, 408, 415, 416, 445, 459, 485

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Integers divisible by 1 more than the number of their divisors. The only prime is 3.

These numbers cannot be very orderly numbers (A167409). - T. D. Noe, Nov 13 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A167409.

Select[Range[1000], Mod[ #, 1 + DivisorSigma[0, # ]] == 0 &]

isok(k) = Mod(k, numdiv(k)+1) == 0; \\ Michel Marcus, Mar 12 2020

Edited by T. D. Noe, Nov 13 2009

A171439 Doubly Orderly Numbers: composite numbers that are orderly for two values of k.

Original entry on oeis.org

393625, 1106861, 2480233, 3166919, 5919509, 6099895, 6440375, 6600349, 8660407, 11407151, 12780523, 14753065, 16900639, 18821573, 21707441, 22671125, 23080813, 23165125, 24924335, 27200929, 28514195, 29947673, 30452005

Offset: 1

Andrew Weimholt, Dec 09 2009

nonn

See A167408 for the definition of Orderly.
All doubly orderly numbers are orderly modulo k=tau(n)+1 and k=tau(n)+3, and are also "very orderly" (Cf. A167409).
Composite numbers appearing in both A167409 and A168003.
No composite number is orderly for more than two values of k, and 11 is the only prime which is orderly for exactly two values of k. 11 does not appear in this sequence as the definition of "doubly orderly" applies only to composite numbers.

			393625 is in the list because it is orderly modulo 17 and 19
.{1,1175,15745,3149,5,125,393625,25,1675,5875,8375,335,47,235,78725,67} == {1,2,3,...,16} mod 17
.{1,393625,1675,5875,5,25,235,78725,47,67,125,335,15745,3149,8375,1175} == {1,2,3,...,16} mod 19

Cf. A167408 - Orderly Numbers
Cf. A167409 - Very Orderly Numbers
Cf. A168003 - Numbers which are orderly modulo tau(n)+3

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A167411 a(n) = the minimal K value for the orderly number A167408(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A055720 Numbers k such that d(k)+1 | k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A171439 Doubly Orderly Numbers: composite numbers that are orderly for two values of k.

Views

Author

Keywords

Comments

Examples

Crossrefs