A097797 -id:A097797 - cp's OEIS frontend

A219601 Number of partitions of n in which no parts are multiples of 6.

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 470, 586, 729, 902, 1113, 1367, 1674, 2042, 2485, 3013, 3645, 4395, 5288, 6344, 7595, 9070, 10809, 12852, 15252, 18062, 21352, 25191, 29671, 34884, 40948, 47985, 56146, 65592

Offset: 0

Arkadiusz Wesolowski, Nov 23 2012

Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012]

			7 = 7
  = 5 + 2
  = 5 + 1 + 1
  = 4 + 3
  = 4 + 2 + 1
  = 4 + 1 + 1 + 1
  = 3 + 3 + 1
  = 3 + 2 + 2
  = 3 + 2 + 1 + 1
  = 3 + 1 + 1 + 1 + 1
  = 2 + 2 + 2 + 1
  = 2 + 2 + 1 + 1 + 1
  = 2 + 1 + 1 + 1 + 1 + 1
  = 1 + 1 + 1 + 1 + 1 + 1 + 1
so a(7) = 14.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Arkadiusz Wesolowski)
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. 15.
Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A097797.
Cf. A261770, A261736, A320608.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

m = 47; f[x_] := (x^6 - 1)/(x - 1); g[x_] := Product[f[x^k], {k, 1, m}]; CoefficientList[Series[g[x], {x, 0, m}], x] (* Arkadiusz Wesolowski, Nov 27 2012 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))

G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k).
a(n) ~ Pi*sqrt(5) * BesselI(1, sqrt(5*(24*n + 5)/6) * Pi/6) / (3*sqrt(24*n + 5)) ~ exp(Pi*sqrt(5*n)/3) * 5^(1/4) / (12 * n^(3/4)) * (1 + (5^(3/2)*Pi/144 - 9/(8*Pi*sqrt(5))) / sqrt(n) + (125*Pi^2/41472 - 27/(128*Pi^2) - 25/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A097798 Number of partitions of n into abundant numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 0, 4, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 8, 0, 5, 0, 2, 0, 14, 0, 4, 0, 4, 0, 14, 0, 8, 0, 5, 0, 23, 0, 9, 0, 9, 0, 26, 0, 18, 0, 9, 0, 38, 0, 16, 0, 17, 0, 46, 0, 29, 0, 19, 0, 65, 0, 32, 0

Offset: 0

Reinhard Zumkeller, Aug 25 2004

nonn

n = 977 = 945 + 32 is the first prime for which sequence obtains a nonzero value, as a(977) = a(32) = 1. 945 is the first term in A005231. - Antti Karttunen, Sep 06 2018

a(n) = 0 for 496 values of n, the largest of which is 991 (see A283550). - David A. Corneth, Sep 08 2018

David A. Corneth, Table of n, a(n) for n = 0..10000 (a(1) through a(532) by Antti Karttunen)
David A. Corneth, PARI program
Eric Weisstein's World of Mathematics, Abundant Number
Eric Weisstein's World of Mathematics, Partition

Cf. A005101, A005231, A000041, A097800, A097797, A097796, A097795.

Cf. also A066874, A282568, A283550.

v:=[n:n in [1..100]| SumOfDivisors(n) gt 2*n]; [#RestrictedPartitions(n,Set(v)): n in [0..100]]; // Marius A. Burtea, Aug 02 2019

n = 100; d = Select[Range[n], DivisorSigma[1, #] > 2 # &]; CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]} ], {x, 0, n}], x] (* Amiram Eldar, Aug 02 2019 *)

abundants_up_to_reversed(n) = { my(s = Set([])); for(k=1,n,if(sigma(k)>(2*k),s = setunion([k],s))); vecsort(s, ,4); };
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A097798(n) = partitions_into(n,abundants_up_to_reversed(n)); \\ Antti Karttunen, Sep 06 2018

PARI
```
\\ see Corneth link
```

a(0) = 1 prepended by David A. Corneth, Sep 08 2018

A097799 Number of partitions of the n-th deficient number into deficient numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 14, 20, 27, 37, 49, 85, 111, 143, 184, 234, 374, 585, 727, 899, 1360, 1664, 2028, 2985, 4343, 5218, 6252, 7474, 8913, 12591, 14918, 17639, 24519, 33836, 39646, 46377, 54165, 63162, 85507, 99276, 115103, 133276, 154116, 205308, 272190

Offset: 1

Reinhard Zumkeller, Aug 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deficient Number
Eric Weisstein's World of Mathematics, Partition

Cf. A005100, A000041, A097797, A097800.

n = 60; d = Select[Range[n], DivisorSigma[1, #] < 2 # &]; CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]}], {x, 0, n}], x][[d + 1]] (* Amiram Eldar, Aug 02 2019 *)

A282566 Number of compositions (ordered partitions) of n into deficient numbers (A005100).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 62, 123, 244, 484, 960, 1904, 3777, 7492, 14861, 29478, 58472, 115984, 230064, 456350, 905208, 1795554, 3561628, 7064780, 14013568, 27797058, 55137735, 109370201, 216944729, 430327593, 853589936, 1693165371, 3358531834, 6661922265, 13214467050, 26211974934, 51993593638, 103133540536

Offset: 0

Ilya Gutkovskiy, Feb 18 2017

nonn

			a(3) = 4 because we have [3], [2, 1], [1, 2] and [1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Deficient Number
Index entries for sequences related to compositions

Cf. A005100, A097797.

nmax = 38; CoefficientList[Series[1/(1 - Sum[Boole[DivisorSigma[1, k] < 2 k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Vec(1/(1 - sum(k=1, 38, (sigma(k)<2*k)*x^k)) + O(x^39)) \\ Indranil Ghosh, Mar 15 2017

G.f.: 1/(1 - Sum_{k>=1} x^A005100(k)).

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A219601 Number of partitions of n in which no parts are multiples of 6.

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A097798 Number of partitions of n into abundant numbers.

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A097799 Number of partitions of the n-th deficient number into deficient numbers.

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A282566 Number of compositions (ordered partitions) of n into deficient numbers (A005100).

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Mathematica

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Formula