A377812 Number of quadruples of positive integers (x,y,a,b) such that a < b, gcd(a,b) = gcd(x,y) = 1 and a*x + b*y = n.

0, 0, 1, 2, 5, 4, 11, 9, 15, 12, 27, 14, 37, 22, 32, 31, 59, 26, 71, 38, 58, 48, 97, 42, 99, 62, 93, 68, 141, 48, 157, 91, 120, 94, 150, 78, 207, 112, 154, 108, 241, 84, 259, 138, 170, 150, 295, 116, 289, 144, 232, 178, 353, 136, 304, 188, 274, 210, 413, 132

Offset: 1

Author

Anshveer Bindra, Nov 08 2024

Keywords

• nonn

Comments

Number of partitions of n into parts with exactly two different sizes, the sizes being relatively prime and also the multiplicities of the two part sizes being relatively prime. - Andrew Howroyd, Nov 10 2024

Crossrefs

Cf. A002133, A274108.

Programs

PARI

a(n)={sum(b=2, n-1, sum(y=1, (n-1)\b, my(s=n-b*y); sumdiv(s, a, aAndrew Howroyd, Nov 10 2024

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(dirmul(u,u), vector(#v, n, v[n]+numdiv(n)-sigma(n))/2)} \\ Andrew Howroyd, Nov 10 2024

Python

def a(n):
    count = 0
    for a in range(1, n+1):
        for b in range(a + 1, n+1):
            if gcd(a, b) == 1:
                for x in range(1, n+1):
                    for y in range(1, n+1):
                        if gcd(x, y) == 1 and a * x + b * y == n:
                            count += 1
    return count
print([a(n) for n in range(1,21)])

Python

from math import gcd
from sympy import divisors
def A377812(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 A274108. - Andrew Howroyd, Nov 10 2024

Extensions

a(21) onwards from Andrew Howroyd, Nov 10 2024