A243935 -id:A243935 - cp's OEIS frontend

A001560 Numbers with an even number of partitions.

2, 8, 9, 10, 11, 15, 19, 21, 22, 25, 26, 27, 28, 30, 31, 34, 40, 42, 45, 46, 47, 50, 55, 57, 58, 59, 62, 64, 65, 66, 70, 74, 75, 78, 79, 80, 84, 86, 94, 96, 97, 98, 100, 101, 103, 106, 108, 109, 110, 112, 113, 116, 117, 120, 122, 124, 125, 126, 128, 129, 130, 131

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).
P. A. MacMahon, The parity of p(n), the number of partitions of n, when n <= 1000, J. London Math. Soc., 1 (1926), 225-226.
T. R. Parkin and D. Shanks, On the distribution of parity in the partition function, Math. Comp., 21 (1967), 466-480.

Cf. A052001, A052002, A000041, A243935.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]; Table[f[2, k], {k, 0, 1}] (* Clark Kimberling, Jan 05 2014 *)

is(n)=numbpart(n)%2==0 \\ Charles R Greathouse IV, Apr 08 2015

A237278 Numbers k such that A000041(k) == 0 (mod 4).

Original entry on oeis.org

11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, 106, 108, 109, 112, 113, 116, 117, 120, 122, 124, 125, 126, 128, 130, 131, 133, 135, 136, 137, 141, 142, 149, 153, 154, 171, 179, 180, 187, 191, 200, 205, 206, 230, 231, 236

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(11) = 56 == 0 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237279, A237280, A237281, A243935 (see crossrefs).

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

is(n)=numbpart(n)%4==0 \\ Charles R Greathouse IV, Apr 08 2015

A083214 Numbers k for which 3 | p(k), where p(k) = A000041(k) is the k-th partition number.

Original entry on oeis.org

3, 7, 9, 10, 14, 16, 17, 20, 21, 22, 24, 26, 30, 32, 33, 35, 39, 40, 41, 43, 46, 48, 51, 52, 53, 57, 61, 63, 68, 70, 71, 75, 80, 88, 97, 102, 104, 106, 107, 111, 115, 124, 125, 129, 133, 138, 142, 147, 151, 160, 162, 163, 164, 169, 173, 178, 180, 181, 189, 191, 193

Offset: 1

Jon Perry, Jun 01 2003

nonn

			A000041(7)=15=0 mod 3.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000041, A087183, A243935.

Select[Range[250],Mod[PartitionsP[ # ],3]==0&] (* Zak Seidov, Apr 03 2007 *)

{ v=[1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604,6842,8349,10143,12310,14883,17977,21637,26015,31185,37338,44583,53174,63261,75175,89134]; for (i=2,length(v)-1,if (v[i]%3==0,print1(i-1","))) }

for(n=1,300,if(polcoeff(1/eta(x)+O(x^(n+1)),n)%3==0,print1(n,","))) \\ Benoit Cloitre, Oct 06 2005

is(n)=numbpart(n)%3==0 \\ Charles R Greathouse IV, Apr 08 2015

Conjecture : a(n) = 3n + o(n). - Benoit Cloitre, Oct 06 2005

A000041(a(n)) = A087183(n). - Zak Seidov, Apr 03 2007

More terms from Benoit Cloitre, Oct 06 2005

A027827 Values of k for which 11 divides A000041(k).

Original entry on oeis.org

6, 8, 12, 15, 16, 17, 18, 20, 21, 25, 28, 29, 31, 32, 35, 37, 38, 39, 41, 42, 43, 49, 50, 51, 52, 54, 56, 58, 59, 61, 62, 64, 72, 74, 75, 78, 81, 82, 83, 84, 85, 87, 94, 96, 98, 104, 105, 107, 108, 109, 116, 117, 118, 119, 125, 127, 128, 129, 130, 131, 138, 140, 148

Offset: 1

R. K. Guy

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000041, A243935.

Select[Range[200], Divisible[PartitionsP[#], 11]&] (* Jean-François Alcover, Dec 12 2016 *)

\ps200 for(n=0,180,if(polcoeff(1/eta(x),n,x)%11==0,print1(n,",")))

is(n)=numbpart(n)%11==0 \\ Charles R Greathouse IV, Apr 08 2015

More terms from Benoit Cloitre, Jun 24 2002

Offset corrected by Amiram Eldar, May 21 2022

A035700 Numbers k such that the number of partitions of k, A000041(k), is a multiple of 12.

Original entry on oeis.org

21, 26, 30, 70, 75, 80, 106, 124, 125, 133, 142, 180, 191, 200, 231, 268, 278, 297, 298, 322, 336, 339, 340, 342, 350, 351, 353, 358, 365, 374, 412, 415, 449, 465, 494, 501, 531, 548, 550, 570, 579, 580, 602, 632, 645, 648, 649, 657, 663, 674, 679, 699

Offset: 1

Olivier Gérard

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000041, A035701, A243935 (see crossrefs).

Select[Range[1000],Mod[PartitionsP[#],12]==0&] (* Harvey P. Dale, May 10 2019 *)

is(n)=numbpart(n)%12==0 \\ Charles R Greathouse IV, Apr 08 2015

A071750 Numbers k such that 13 divides p(k), the k-th partition number, A000041(k).

Original entry on oeis.org

28, 62, 84, 94, 129, 136, 173, 180, 197, 213, 219, 226, 227, 237, 240, 264, 294, 311, 318, 326, 335, 338, 357, 358, 389, 418, 453, 458, 473, 482, 484, 486, 508, 529, 538, 542, 562, 600, 635, 644, 668, 670, 684, 697, 699, 713, 714, 727, 742, 747, 751, 778

Offset: 1

Benoit Cloitre, Jun 24 2002

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000041, A027827, A243935 (see crossrefs).

Select[ Range[800], Mod[ PartitionsP[ # ], 13] == 0 &]

\ps800 for(n=0,600,if(polcoeff(1/eta(x),n,x)%13==0,print1(n,",")))

is(n)=numbpart(n)%13==0 \\ Charles R Greathouse IV, Apr 08 2015

Edited by Robert G. Wilson v, Jun 27 2002

A243936 Numbers m such that 7 divides A000041(m).

Original entry on oeis.org

5, 10, 11, 12, 16, 18, 19, 24, 26, 27, 33, 37, 39, 40, 41, 47, 48, 52, 53, 54, 55, 61, 68, 75, 76, 82, 83, 89, 96, 97, 103, 110, 111, 117, 124, 125, 131, 138, 140, 145, 147, 152, 159, 166, 170, 173, 177, 180, 187, 191, 194, 201, 208, 213, 215, 222, 225, 229, 232

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), A243935 (k=5), this sequence (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..250] | IsZero(NumberOfPartitions(n) mod 7)];

Select[Range[250], Mod[PartitionsP[#], 7] == 0 &]

is(n)=numbpart(n)%7==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..250) if mod(A000041(n),7) == 0]

cp's OEIS Frontend

A001560 Numbers with an even number of partitions.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

A237278 Numbers k such that A000041(k) == 0 (mod 4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A083214 Numbers k for which 3 | p(k), where p(k) = A000041(k) is the k-th partition number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A027827 Values of k for which 11 divides A000041(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A035700 Numbers k such that the number of partitions of k, A000041(k), is a multiple of 12.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A071750 Numbers k such that 13 divides p(k), the k-th partition number, A000041(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A243936 Numbers m such that 7 divides A000041(m).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage