A359937 -id:A359937 - cp's OEIS frontend

A359936 Expansion of Product_{k>=0} (1 - x^(k^2+1)) in powers of x.

1, -1, -1, 1, 0, -1, 1, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -2, 2, 1, -1, 0, 1, -1, -1, 1, -1, 2, 0, -2, 1, 1, -2, 0, 2, -1, 1, -2, 0, 2, -1, -1, 2, 1, -3, 0, 1, 1, -2, 0, 1, 0, 0, -1, 3, 0, -3, 0, 2, -2, 1, 1, -2, 2, -2, -2, 4, 2, -4, 1, 2, -3, -1, 1, 2, 1, -1, -3, 2

Offset: 0

Seiichi Manyama, Jan 19 2023

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

Cf. A002522, A081362, A276516, A284312, A284313, A284314, A284585, A284499, A357911, A359937.

my(N=100, x='x+O('x^N)); Vec(prod(k=0, sqrtint(N), 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} A359937(k) * a(n-k).

A359967 a(n) = Sum_{d|n, d+1 is square} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 8, 3, 0, 0, 3, 0, 0, 18, 8, 0, 3, 0, 0, 3, 0, 0, 35, 0, 0, 3, 0, 0, 18, 0, 8, 3, 0, 35, 3, 0, 0, 3, 8, 0, 3, 0, 0, 18, 0, 0, 83, 0, 0, 3, 0, 0, 3, 0, 8, 3, 0, 0, 18, 0, 0, 66, 8, 0, 3, 0, 0, 3, 35, 0, 35, 0, 0, 18, 0, 0, 3, 0, 88, 3, 0, 0, 3, 0, 0

Offset: 1

Seiichi Manyama, Jan 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^7

Cf. A005563, A035316, A179952, A359937.

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)); concat([0, 0], Vec(sum(k=2, sqrtint(N+1), (k^2-1)*x^(k^2-1)/(1-x^(k^2-1)))))

G.f.: Sum_{k>=2} (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

A359943 a(n) = Sum_{d|n, d-1 is cube} d.

Original entry on oeis.org

1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 31, 1, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 3, 1, 3, 1, 3, 1, 12, 1, 31, 1, 3, 1, 3, 1, 3, 10, 3, 66, 3, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 3, 1, 3, 10, 3, 1, 31, 1, 3

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

Cf. A001093, A113061, A359937, A359942, A359944.

a[n_] := DivisorSum[n, # &, IntegerQ[Surd[#-1, 3]] &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

PARI
```
a(n) = sumdiv(n, d, ispower(d-1, 3)*d);
```

G.f.: Sum_{k>=0} (k^3+1) * x^(k^3+1)/(1 - x^(k^3+1)).

A359938 Number of divisors d of n such that d-1 is square.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 2, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 2, 1, 5, 2, 3, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 1, 2, 1, 3, 2, 2, 1, 3, 1, 4, 1, 3, 1, 2, 3, 2, 1, 2, 1, 4

Offset: 1

Seiichi Manyama, Jan 19 2023

nonn

The Cartesian equation for the Witch of Agnesi is given as y = 8*k^3/(x^2+4*k^2). If we set x = n, then a(n)-1 is the number of positive integer solutions in k such that y is a positive integer. Let d = m^2+1 be a divisor of n, then k = m*n/2 is a solution. - Thomas Scheuerle, Aug 07 2024

Cf. A002522, A046951, A113319, A359937, A359944.

nmax = 100; Rest[CoefficientList[Series[Sum[x^(k^2 + 1)/(1 - x^(k^2 + 1)), {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 19 2023 *)
Table[Sum[Floor[n/(k^2 + 1)] - Floor[(n-1)/(k^2 + 1)], {k, 0, Sqrt[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 19 2023 *)

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

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

G.f.: Sum_{k>=0} x^(k^2+1)/(1 - x^(k^2+1)).
From Vaclav Kotesovec, Jan 19 2023: (Start)
a(n) = Sum_{k=0..n} (floor(n/(k^2 + 1)) - floor((n-1)/(k^2 + 1))).
Sum_{k=1..n} a(k) = Sum_{k=0..n} floor(n/(k^2 + 1)).
Sum_{k=1..n} a(k) ~ c*n, where c = A113319 = (1 + Pi*coth(Pi))/2 = 2.0766... (End)

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A359936 Expansion of Product_{k>=0} (1 - x^(k^2+1)) in powers of x.

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A359967 a(n) = Sum_{d|n, d+1 is square} d.

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A359943 a(n) = Sum_{d|n, d-1 is cube} d.

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A359938 Number of divisors d of n such that d-1 is square.

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