A147787 -id:A147787 - cp's OEIS frontend

A147783 Number of partitions of n into parts divisible by 2 or 5.

1, 0, 1, 0, 2, 1, 3, 1, 5, 2, 8, 3, 12, 5, 17, 9, 25, 13, 35, 19, 51, 28, 69, 40, 96, 59, 129, 81, 175, 113, 236, 154, 313, 210, 412, 286, 542, 381, 705, 506, 921, 668, 1185, 875, 1525, 1148, 1948, 1485, 2485, 1918, 3157, 2462, 3990, 3150

Offset: 0

Alexander E. Holroyd (holroyd at math.ubc.ca)

nonn

Also number of partitions of n with no part and no difference between two parts equal to 1 or 3.

Also number of partitions of n with no part appearing 1 or 3 times.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
G. E. Andrews, A Generalization of a Partition Theorem of MacMahon, J. Combin. Theory, 3 (1967) 100-101.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, arXiv:0706.2282 [math.CO], 2007.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.

Cf. A007690, A147784, A147785, A147786, A147787.

nmax = 60; CoefficientList[Series[Product[(1 + x^(5*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)

G.f.: Product_{k>=1} (1-x^(10k))/(1-x^(2k))/(1-x^(5k)).

a(n) ~ exp(sqrt(2*n/5)*Pi)/(4*sqrt(5)*n). - Vaclav Kotesovec, Sep 23 2015

A147784 Number of partitions of n into parts divisible by 3 or 4.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 1, 2, 3, 2, 2, 7, 3, 4, 9, 9, 6, 15, 11, 15, 21, 19, 19, 39, 27, 32, 51, 51, 45, 78, 67, 82, 107, 104, 108, 172, 143, 165, 226, 232, 226, 328, 306, 356, 441, 446, 470, 655, 601, 677, 857, 891, 908, 1197, 1169, 1325, 1582

Offset: 0

Alexander E. Holroyd (holroyd at math.ubc.ca)

nonn

Also number of partitions of n with no part and no difference between two parts equal to 1,2 or 5.

Also number of partitions of n with no part appearing 1,2 or 5 times.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, arXiv:0706.2282 [math.CO], 2007.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.

Cf. A007690, A147783, A147785, A147786, A147787.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))*(1 + x^(6*k))/(1 - x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)

G.f.: Product_{k>=1} (1-x^(12k))/(1-x^(3k))/(1-x^(4k)).

a(n) ~ exp(sqrt(n/3)*Pi)/(4*sqrt(6)*n). - Vaclav Kotesovec, Sep 23 2015

A147785 Number of partitions of n into parts divisible by 3 or 5.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 0, 1, 3, 2, 2, 5, 2, 3, 9, 4, 5, 13, 6, 11, 19, 10, 15, 28, 19, 23, 40, 27, 34, 63, 40, 50, 85, 59, 79, 121, 85, 109, 166, 132, 155, 230, 180, 216, 325, 255, 300, 436, 351, 429, 588, 485, 576, 789, 680, 784, 1050, 912, 1053, 1421, 1228

Offset: 0

Alexander E. Holroyd (holroyd at math.ubc.ca)

nonn

Also number of partitions of n with no part and no difference between two parts equal to 1,2,4 or 7.

Also number of partitions of n with no part appearing 1,2,4 or 7 times.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, arXiv:0706.2282 [math.CO], 2007.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 17.

Cf. A007690, A147783, A147784, A147786, A147787.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 3)=0 or irem(d, 5)=0, d, 0),
           d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Dec 02 2016

nmax = 60; CoefficientList[Series[Product[(1-x^(15*k))/((1-x^(3*k))*(1-x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)

G.f.: Product_{k>=1} (1-x^(15k)) / ((1-x^(3k))*(1-x^(5k))).

a(n) ~ sqrt(7/5) * exp(sqrt(14*n/5)*Pi/3) / (12*n). - Vaclav Kotesovec, Sep 23 2015

A147786 Number of partitions of n into parts divisible by 4 or 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 3, 2, 2, 3, 5, 3, 4, 3, 11, 5, 6, 6, 15, 13, 10, 9, 23, 17, 23, 15, 34, 27, 31, 33, 50, 40, 48, 45, 86, 60, 71, 69, 116, 106, 105, 102, 169, 144, 176, 150, 237, 211, 240, 248, 335, 299, 347, 338, 506, 425, 487, 487, 681

Offset: 0

Alexander E. Holroyd (holroyd at math.ubc.ca)

nonn

Also number of partitions of n with no part and no difference between two parts equal to 1,2,3,6,7 or 11.

Also number of partitions of n with no part appearing 1,2,3,6,7 or 11 times.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, arXiv:0706.2282 [math.CO], 2007.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.

Cf. A007690, A147783, A147784, A147785, A147787.

nmax = 60; CoefficientList[Series[Product[(1 + x^(5*k))*(1 + x^(10*k))/(1 - x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)

G.f.: Product_{k>=1} (1-x^(20k))/(1-x^(4k))/(1-x^(5k)).

a(n) ~ exp(2*Pi*sqrt(n/15))/(2*sqrt(30)*n). - Vaclav Kotesovec, Sep 23 2015

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A147783 Number of partitions of n into parts divisible by 2 or 5.

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A147784 Number of partitions of n into parts divisible by 3 or 4.

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A147785 Number of partitions of n into parts divisible by 3 or 5.

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A147786 Number of partitions of n into parts divisible by 4 or 5.

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