A359967 -id:A359967 - cp's OEIS frontend

A359937 a(n) = Sum_{d|n, d-1 is square} d.

1, 3, 1, 3, 6, 3, 1, 3, 1, 18, 1, 3, 1, 3, 6, 3, 18, 3, 1, 18, 1, 3, 1, 3, 6, 29, 1, 3, 1, 18, 1, 3, 1, 20, 6, 3, 38, 3, 1, 18, 1, 3, 1, 3, 6, 3, 1, 3, 1, 68, 18, 29, 1, 3, 6, 3, 1, 3, 1, 18, 1, 3, 1, 3, 71, 3, 1, 20, 1, 18, 1, 3, 1, 40, 6, 3, 1, 29, 1, 18, 1, 85, 1, 3

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002522, A035316, A359967.

Table[Sum[If[IntegerQ[Sqrt[d-1]], d, 0], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 21 2023 *)

PARI
```
a(n) = sumdiv(n, d, issquare(d-1)*d);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=0, sqrtint(N), (k^2+1)*x^(k^2+1)/(1-x^(k^2+1))))

G.f.: Sum_{k>=0} (k^2+1) * x^(k^2+1)/(1 - x^(k^2+1)).

Sum_{k=1..n} a(k) ~ zeta(3/2)*n^(3/2)/3. - Vaclav Kotesovec, Jan 21 2023

A359966 Expansion of Product_{k>=2} (1 - x^(k^2-1)) in powers of x.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 1, 1, 0, 0, -1, 1, 0, 0, -1, -1, -1, 0, 2, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1, 0, -1, -1, 0, 1, 1, 0, -2, -1, 1, 3

Offset: 0

Seiichi Manyama, Jan 20 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A005563, A276516, A359936, A359967.

my(N=100, x='x+O('x^N)); Vec(prod(k=2, sqrtint(N+1), 1-x^(k^2-1)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, sumdiv(j, d, issquare(d+1)*d)*v[i-j+1])/i); v;

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A359967(k) * a(n-k).

cp's OEIS Frontend

A359937 a(n) = Sum_{d|n, d-1 is square} d.

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A359966 Expansion of Product_{k>=2} (1 - x^(k^2-1)) in powers of x.

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Author

Keywords

Links

Crossrefs

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PARI

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Formula