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

1,4

Links

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.

Formula

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

Example

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

Prog

(PARI) 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
(Python)
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 a<b and gcd(a, b)==1) # Chai Wah Wu, Dec 11 2024

Crossrefs

Row sums of triangle in A274109.

Cf. A002133, A117955, A117956, A216665.

Sequence in context: A194119 A084721 A273041 * A368616 A138316 A377043

Adjacent sequences: A274105 A274106 A274107 * A274109 A274110 A274111

Keyword

nonn,look

Author

N. J. A. Sloane, Jun 17 2016

Extensions

More terms from Alois P. Heinz, Jun 23 2016

Status

approved