A216665 -id:A216665 - cp's OEIS frontend

A274108 Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime.

0, 0, 1, 2, 5, 5, 11, 11, 16, 17, 27, 21, 37, 33, 38, 42, 59, 46, 71, 57, 70, 75, 97, 72, 104, 99, 109, 103, 141, 102, 157, 133, 148, 153, 166, 140, 207, 183, 192, 174, 241, 180, 259, 215, 223, 247, 295, 219, 300, 260, 292, 279, 353, 275, 336, 300, 346, 351

Offset: 1

N. J. A. Sloane, Jun 17 2016

			Explanation of a(3)-a(6):
n=3: 21
n=4: 31, 211
n=5: 41, 32, 311, 221, 2111
n=6: 51, 411, 3111, 2211, 21111

Alois P. Heinz, Table of n, a(n) for n = 1..10000
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.

Row sums of triangle in A274109.

Cf. A002133, A117955, A117956, A216665.

seq(n)={my(v=Vec(sum(k=1, n-1, numdiv(k)*x^k, O(x^n))^2, -n), u=vector(n, n, moebius(n))); dirmul(u, vector(#v, n, v[n]+numdiv(n)-sigma(n))/2)} \\ Andrew Howroyd, Nov 10 2024

from math import gcd
from sympy import divisors
def A274108(n): return sum(1 for ax in range(1,n-1) for a in divisors(ax,generator=True) for b in divisors(n-ax,generator=True) if aChai Wah Wu, Dec 11 2024

Formula

Moebius transform of A002133. - Andrew Howroyd, Nov 10 2024

Extensions

More terms from Alois P. Heinz, Jun 23 2016

A216669 Total number of parts in all partitions of n into 2 sizes of parts (A002133).

Original entry on oeis.org

2, 5, 14, 20, 39, 52, 74, 102, 134, 158, 208, 259, 284, 361, 409, 488, 538, 634, 678, 838, 857, 1006, 1038, 1270, 1264, 1495, 1500, 1776, 1761, 2084, 2062, 2443, 2300, 2795, 2680, 3194, 3076, 3544, 3403, 4080, 3804, 4518, 4282, 5037, 4673, 5626, 5127, 6088

Offset: 3

Geoffrey Critzer, Sep 13 2012

nonn

Also column k=2 of A255768. - Omar E. Pol, Jul 26 2015

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A002133, A216665, A255768.

nn=40;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}], {i,1,nn}]; CoefficientList[Series[D[ss,y]/.y->1,{x,0,nn}],x]

a(n) = Sum_{k=2..n-1} k * A216665(n,k).

A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 3, 2, 3, 2, 2, 1, 2, 2, 4, 1, 3, 2, 2, 1, 5, 5, 3, 4, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 1, 6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1, 3, 3, 5, 3, 4, 1, 4, 2, 3, 2, 2, 1, 4, 4, 3, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1, 4, 4, 5, 3, 4, 3, 3, 2, 4, 2, 3, 2, 2, 1, 8, 8, 5, 7, 3, 5, 3, 4, 2, 4, 2, 3, 2, 2, 1, 3, 3, 5, 2, 5, 2, 4, 2, 4, 2, 4, 2, 3, 2, 2, 1, 9, 9, 6, 7, 3, 7, 3, 4, 3, 4, 2, 4, 2, 3, 2, 2, 1

Offset: 3

N. J. A. Sloane, Jun 17 2016

			Triangle T(n,k) (with columns n >= 3 and k >= 2) begins as follows:
  1;
  1, 1;
  2, 2, 1;
  1, 1, 2, 1;
  3, 3, 2, 2, 1;
  2, 2, 2, 2, 2, 1;
  3, 3, 2, 3, 2, 2, 1;
  2, 2, 4, 1, 3, 2, 2, 1;
  5, 5, 3, 4, 2, 3, 2, 2, 1;
  2, 2, 2, 2, 3, 2, 3, 2, 2, 1;
  6, 6, 4, 5, 2, 4, 2, 3, 2, 2, 1;
  3, 3, 5, 3, 4, 1, 4, 2, 3, 2, 2, 1;
  4, 4, 3, 3, 4, 4, 2, 4, 2, 3, 2, 2, 1;
  ...

N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Elemente der Mathematik 72(2) (2017), 66-74.

Row sums give A274108.

Cf. A002133, A117955, A117956, A216665.

cp's OEIS Frontend

A274108 Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime.

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A216669 Total number of parts in all partitions of n into 2 sizes of parts (A002133).

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A274109 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts with exactly two different sizes, the sizes being relatively prime (n >= 3, 2 <= k <= n-1).

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