A076752 -id:A076752 - cp's OEIS frontend

A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2k^2)) / (1 - x^(2k^2))^2.

1, 0, 3, 1, 5, 0, 7, 0, 10, 0, 11, 3, 13, 0, 15, 1, 17, 0, 19, 5, 21, 0, 23, 0, 26, 0, 30, 7, 29, 0, 31, 0, 33, 0, 35, 10, 37, 0, 39, 0, 41, 0, 43, 11, 50, 0, 47, 3, 50, 0, 51, 13, 53, 0, 55, 0, 57, 0, 59, 15, 61, 0, 70, 1, 65, 0, 67, 17, 69, 0, 71, 0, 73, 0, 78, 19, 77, 0, 79, 5, 91

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

Sum of odd divisors d of n such that n/d is square.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000593, A010052, A035316, A036554 (positions of 0's), A056911 (fixed points), A076752, A193356, A328372.

a:=[];for n in [1..81] do  v:=[d:d in Divisors(n)| IsOdd(d) and IsSquare(n div d)]; if #v ne 0  then Append(~a,&+v); else Append(~a,0); end if; end for; a; // Marius A. Burtea, Oct 14 2019

nmax = 81; CoefficientList[Series[Sum[x^(k^2) (1 + x^(2 k^2))/(1 - x^(2 k^2))^2, {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, OddQ[#] && IntegerQ[(n/#)^(1/2)] &], {n, 1, 81}]
f[p_, e_] := If[p == 2, Boole @ EvenQ[e], If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), (p^(e + 2) - p)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 16 2020 *)

a(n) = sumdiv(n, d, if ((d%2) && issquare(n/d), d)); \\ Michel Marcus, Oct 14 2019

G.f.: Sum_{k>=1} (2*k - 1) * (theta_3(x^(2*k - 1)) - 1) / 2.
G.f.: Sum_{i>=1} Sum_{j>=1} phi(i) * x^(i*j^2) / (1 + x^(i*j^2)).
Dirichlet g.f.: (1 - 2^(1 - s)) * zeta(s-1) * zeta(2*s).
a(n) = Sum_{d|n} A193356(d) * A010052(n/d).
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 360. - Vaclav Kotesovec, Oct 14 2019
Multiplicative with a(2^e) = 0 if e is odd, and 1 if e is even, and for p > 2, a(p^e) = (p^(e + 2) - p)/(p^2 - 1) if e is odd, and (p^(e + 2) - 1)/(p^2 - 1) if e is even. - Amiram Eldar, Oct 16 2020

A328271 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3.

Original entry on oeis.org

1, 4, 9, 17, 25, 36, 49, 68, 82, 100, 121, 153, 169, 196, 225, 273, 289, 328, 361, 425, 441, 484, 529, 612, 626, 676, 738, 833, 841, 900, 961, 1092, 1089, 1156, 1225, 1394, 1369, 1444, 1521, 1700, 1681, 1764, 1849, 2057, 2050, 2116, 2209, 2457, 2402, 2504, 2601, 2873, 2809, 2952, 3025

Offset: 1

Ilya Gutkovskiy, Oct 10 2019

Sum of squares of divisors d of n such that n/d is square.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000290, A001157, A010052, A076752, A076792.

[&+[d^2:d in Divisors(n)| IsSquare(n div d)]:n in [1..55]]; // Marius A. Burtea, Oct 10 2019

a:= n-> add((n/d)^2, d=select(issqr, numtheory[divisors](n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 11 2019

nmax = 55; CoefficientList[Series[Sum[x^(k^2) (1 + x^(k^2))/(1 - x^(k^2))^3, {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^2 &, IntegerQ[Sqrt[n/#]] &], {n, 1, 55}]

a(n) = sumdiv(n, d, if (issquare(n/d), d^2)); \\ Michel Marcus, Oct 12 2019

G.f.: Sum_{k>=1} k^2 * (theta_3(x^k) - 1)/2.
Dirichlet g.f.: zeta(2*s) * zeta(s-2).
a(n) = Sum_{d|n} A010052(n/d) * d^2.
a(n) = Sum_{d|n} |A076792(d)|.
a(p) = p^2, where p is prime.
Sum_{k=1..n} a(k) ~ Pi^6 * n^3 / 2835. - Vaclav Kotesovec, Oct 11 2019
Multiplicative with a(p^e) = Sum_{i=0..floor(e/2)} p^(2*e-4*i) for prime p, i.e., a(p^(2*e)) = (p^(4*e+4)-1)/(p^4-1) and a(p^(2*e+1)) = p^2 * (p^(4*e+4)-1)/(p^4-1) for prime p. - Werner Schulte, Jul 24 2021

A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 7, 7, 8, 8, 10, 10, 10, 10, 12, 13, 13, 14, 16, 16, 16, 16, 19, 19, 19, 19, 24, 24, 24, 24, 28, 28, 28, 28, 32, 34, 34, 34, 39, 40, 41, 41, 46, 46, 48, 48, 53, 53, 53, 53, 58, 58, 58, 60, 67, 67, 67, 67, 74, 74, 74, 74, 84, 84, 84, 85, 93, 93, 93, 93

Offset: 1

Ilya Gutkovskiy, Jan 28 2025

nonn

Cf. A003238, A076752, A167865, A307779, A317240.

a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      add(`if`(issqr(d), a((n-1)/d), 0), d=divisors(n-1)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 28 2025

a[1] = 1; a[n_] := a[n] = DivisorSum[n - 1, a[(n - 1)/#] &, IntegerQ[Sqrt[#]] &]; Table[a[n], {n, 1, 80}]
nmax = 80; A[] = 0; Do[A[x] = x (1 + Sum[A[x^(k^2)], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x^4) + A(x^9) + ... + A(x^(k^2)) + ...).

cp's OEIS Frontend

A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2k^2)) / (1 - x^(2k^2))^2.

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A328271 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3.

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A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

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cp's OEIS Frontend

A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2*k^2)) / (1 - x^(2*k^2))^2.

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A328271 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3.

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Magma

Maple

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A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

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Maple

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A328373 Expansion of Sum_{k>=1} x^(k^2) * (1 + x^(2k^2)) / (1 - x^(2k^2))^2.