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.

A236103 Number of distinct partition numbers dividing n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 1, 3, 4, 2, 1, 3, 1, 3, 3, 4, 1, 3, 2, 2, 2, 3, 1, 6, 1, 2, 3, 2, 3, 3, 1, 2, 2, 3, 1, 5, 1, 4, 4, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 2, 4, 2, 4, 3, 1, 3, 2, 2, 1, 5, 2, 2, 2, 4, 1, 6
Offset: 1

Views

Author

Omar E. Pol, Jan 21 2014

Keywords

Examples

			For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20 and three of them are also partition numbers: 1, 2, 5, so a(20) = 3.
For n = 42 the divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and five of them are also partition numbers: 1, 2, 3, 7, 42, so a(42) = 5.

Links

Crossrefs

Cf. A000041, A001221, A049575, A078506, A085543, A167392, A236102, A236105, A236107, A236108.

Programs

  • Mathematica
    p = {1}; Table[If[n >= Last@p, AppendTo[p, PartitionsP[1 + Length@p]]]; Length@Select[p, Mod[n, #] == 0 &], {n, 90}] (* Giovanni Resta, Jan 22 2014 *)

Formula

From Amiram Eldar, Jan 01 2024: (Start)
a(n) = Sum_{d|n} A167392(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A078506 = 2.510597... . (End)

A236102 Numbers whose divisors are partition numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 77, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557, 74878248419470886233, 1394313503224447816939
Offset: 1

Views

Author

Omar E. Pol, Jan 21 2014

Keywords

Comments

By definition all terms are partition numbers.
All members of A049575 are in this sequence.
Conjecture: the only composite numbers in this sequence are 15, 22, and 77. - Jon E. Schoenfield, Feb 05 2014

Examples

			15 is in the sequence because the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.

Crossrefs

Cf. A000041, A027750, A049575, A236103, A236105, A236107, A236108, A236110, A236111.

Extensions

More terms from Jon E. Schoenfield, Feb 05 2014

A236105 Numbers whose proper divisors are partition numbers.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 202
Offset: 1

Views

Author

Omar E. Pol, Jan 21 2014

Keywords

Comments

All prime numbers are in the sequence.
All terms of A236102 are in this sequence, except the first term.

Examples

			10 is in the sequence because the proper divisors of 10 are 1, 2, 5, which are also partition numbers.

Crossrefs

Cf. A000040, A000041, A236102, A236103, A236107, A236108, A236110.

A236107 Nonprimes whose divisors are partition numbers.

Original entry on oeis.org

1, 15, 22, 77
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2014

Keywords

Comments

By definition all terms are partition numbers.
Conjecture: no terms exist beyond 77. - Jon E. Schoenfield, Feb 05 2014

Examples

			15 is in the sequence because 15 is a nonprime number and the divisors of 15 are 1, 3, 5, 15, which are also partition numbers.

Crossrefs

Cf. A000041, A018252, A038753, A236102, A236103, A236105, A236108, A236110.

Programs

  • Mathematica
    nmax = 1000;
pp = PartitionsP[Range[nmax]];
selQ[n_] := Module[{dd = Divisors[n]}, Intersection[pp, dd] == dd];
Select[Range[nmax], !PrimeQ[#] && selQ[#]&] (* Jean-François Alcover, Apr 09 2020 *)

A236110 Smallest number with the property that exactly n of its divisors are partition numbers.

Original entry on oeis.org

1, 2, 6, 15, 42, 30, 270, 210, 462, 1848, 3696, 11088, 2310, 9240, 18480, 55440, 83160, 166320, 498960, 2494800, 17463600, 331808400, 4418290800
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2014

Keywords

Examples

			a(3) = 6 because 6 is the smallest number with the property that exactly three of its divisors are partition numbers. The divisors of 6 are 1, 2, 3, 6, and 1, 2, 3 are also partition numbers.
a(5) = 42 because 42 is the smallest number with the property that exactly five of its divisors are partition numbers. The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and 1, 2, 3, 7, 42 are members of A000041.

Crossrefs

Cf. A000041, A236102, A236103, A236105, A236107, A236108, A236111.

Extensions

a(12) and a(15)-a(18) from Alois P. Heinz, Jan 22 2014
a(19)-a(22) from Giovanni Resta, Feb 06 2014
a(23) from Amiram Eldar, Jun 23 2023

A236111 Numbers that set a record for the number of divisors that are partition numbers.

Original entry on oeis.org

1, 2, 6, 15, 30, 210, 462, 1848, 2310, 9240, 18480, 55440, 83160, 166320, 498960, 2494800, 17463600, 331808400, 4418290800
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2014

Keywords

Comments

Where records occur in A236103.

Crossrefs

Cf. A000041, A236102, A236103, A236105, A236107, A236108, A236110.

Extensions

More terms from Jon E. Schoenfield, Feb 08 2014
a(18)-a(19) from Amiram Eldar, Jun 23 2023
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.