cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Andrew Weimholt, Nov 03 2009

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • 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[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

Views

Author

Andrew Weimholt, Nov 03 2009

Keywords

Examples

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

Links

Crossrefs

Cf. A167408 - Orderly Numbers.
Cf. A167409 - Very Orderly Numbers ( K = tau(N)+1 ).
Cf. A167410 - Disorderly Numbers - numbers not in A167408.

Programs

  • Mathematica
    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 *)

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

Views

Author

Andrew Weimholt, Nov 03 2009

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

A168003 Orderly numbers (mod tau(n)+3).

Original entry on oeis.org

7, 37, 67, 97, 127, 157, 255, 277, 307, 337, 367, 397, 457, 487, 547, 577, 607, 727, 757, 787, 877, 907, 915, 937, 967, 997, 1087, 1117, 1237, 1245, 1297, 1327, 1447, 1567, 1597, 1627, 1657, 1747, 1777, 1867, 1905, 1987, 2017, 2125, 2137, 2235, 2287, 2347
Offset: 1

Views

Author

T. D. Noe, Nov 16 2009

Keywords

Comments

See A167408 for information about orderly numbers. It appears that when n is in this sequence, then tau(n)+3 must be a prime p such that 2 is not a square mod p (A003629). For each one of those primes, it is possible to find all forms of n that are orderly. In particular, the form n=p^k*q is in this sequence when 2k+5 is in A001122. In that case, we have the congruences p=2+tau(n)/2 and q=1+tau(n)/2 (mod tau(n)+3). When tau(n) is a multiple of 8, then another pair of congruences is p=1+tau(n)/2 and q=2+tau(n)/2 (mod tau(n)+3).

Formula

An exhaustive search over forms of n having a prime value of tau(n)+3 finds that terms of this sequence satisfy the following congruences for tau(n)+3 < 60.
. p with prime p = 2 mod 5
. p^3*q with primes {p,q} == {5,6} mod 11
. p^3*q with primes {p,q} == {6,5} mod 11
. p*q*r with primes {p,q,r} == {3,5,6} mod 11
. p^4*q with primes {p,q} == {7,6} mod 13
. p^7*q with primes {p,q} == {9,10} mod 19
. p^7*q with primes {p,q} == {10,9} mod 19
. p^3*q*r with primes {p,q,r} == {5,9,10} mod 19
. p^3*q*r with primes {p,q,r} == {9,6,10} mod 19
. p^3*q*r with primes {p,q,r} == {10,6,9} mod 19
. p*q*r*s with primes {p,q,r,s} == {5,6,9,10} mod 19
. p^12*q with primes {p,q} == {15,14} mod 29
. p^16*q with primes {p,q} == {19,18} mod 37
. p^4*q*r*s with primes {p,q,r,s} == {14,13,15,22} mod 43
. p^4*q*r*s with primes {p,q,r,s} == {31,22,24,38} mod 43
. p^24*q with primes {p,q} == {27,26} mod 53
. p^4*q^4*r with primes {p,q,r} == {5,27,26} mod 53
. p^27*q with primes {p,q} == {29,30} mod 59
. p^27*q with primes {p,q} == {30,29} mod 59
. p^13*q*r with primes {p,q,r} == {15,29,30} mod 59
. p^13*q*r with primes {p,q,r} == {29,30,36} mod 59
. p^13*q*r with primes {p,q,r} == {30,29,36} mod 59
. p^6*q^3*r with primes {p,q,r} == {29,53,30} mod 59
. p^6*q^3*r with primes {p,q,r} == {30,6,29} mod 59
. p^6*q^3*r with primes {p,q,r} == {48,29,30} mod 59
. p^6*q^3*r with primes {p,q,r} == {48,30,29} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {7,28,30,45} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {15,29,30,36} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {29,30,36,53} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {30,6,29,36} mod 59
. p^6*q*r*s with primes {p,q,r,s} == {48,15,29,30} mod 59
Andrew Weimholt found some of these forms.

Extensions

Comment corrected and congruences mod 43 added by T. D. Noe, Dec 02 2009

A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0, 15017, 1476227, 215233607, 455627, 17327, 150094635296999123, 15884240051, 89813531, 36857, 0
Offset: 1

Views

Author

T. D. Noe, Nov 09 2009

Keywords

Comments

This sequence is the idea of Alonso Del Arte. For n>2, a(n) is conjectured to be the smallest number that is orderly (see A167408) for n-1 values of k. For example, 11 is orderly for k=3 and 9. See A056899 for other primes p that are orderly for two k. It is a conjecture because it is not known whether there are composite numbers that are orderly for more than one value of k.
The terms a(n) for prime n are 0 except when 3^(n-1)+2 is prime. Using A051783, we find the exceptional primes to be n=2, 3, 5, 11, 37, 127, 6959.... For these n, a(n) = 3^(n-1)+2. For any n, it is easy to use the factorization of n to find the forms of numbers that have n divisors. For example, for n=38=2*19, we know that the prime must have the form 2+q*r^18 with q and r prime. The smallest such prime is 2+41*3^18.

Crossrefs

Cf. A066814 (smallest prime p such that p-1 has n divisors)

Programs

  • Mathematica
    nn=25; t=Table[0,{nn}]; Do[p=Prime[n]; k=DivisorSigma[0,p-2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n,2,10^6}]; t

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

Views

Author

Andrew Weimholt, Dec 09 2009

Keywords

Comments

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.

Examples

			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

Crossrefs

Cf. A167408 - Orderly Numbers
Cf. A167409 - Very Orderly Numbers
Cf. A168003 - Numbers which are orderly modulo tau(n)+3
Showing 1-6 of 6 results.

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