A274108 - OEIS

%I #29 Dec 11 2024 21:16:13

%S 0,0,1,2,5,5,11,11,16,17,27,21,37,33,38,42,59,46,71,57,70,75,97,72,

%T 104,99,109,103,141,102,157,133,148,153,166,140,207,183,192,174,241,

%U 180,259,215,223,247,295,219,300,260,292,279,353,275,336,300,346,351

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

%H Alois P. Heinz, <a href="/A274108/b274108.txt">Table of n, a(n) for n = 1..10000</a>

%H N. Benyahia Tani, S. Bouroubi, and O. Kihel, <a href="https://liforce.usthb.dz/sites/default/files/2020-11/article3.pdf">An effective approach for integer partitions using exactly two distinct sizes of parts</a>, Bulletin du Laboratoire 03 (2015), 18-27.

%F Moebius transform of A002133. - _Andrew Howroyd_, Nov 10 2024

%e Explanation of a(3)-a(6):

%e n=3: 21

%e n=4: 31, 211

%e n=5: 41, 32, 311, 221, 2111

%e n=6: 51, 411, 3111, 2211, 21111

%o (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

%o (Python)

%o from math import gcd

%o from sympy import divisors

%o 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

%Y Row sums of triangle in A274109.

%Y Cf. A002133, A117955, A117956, A216665.

%K nonn,look

%O 1,4

%A _N. J. A. Sloane_, Jun 17 2016

%E More terms from _Alois P. Heinz_, Jun 23 2016