A347090 -id:A347090 - cp's OEIS frontend

A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

1, 3, 4, 8, 6, 12, 8, 18, 14, 18, 12, 32, 14, 24, 24, 39, 18, 42, 20, 48, 32, 36, 24, 72, 32, 42, 44, 64, 30, 72, 32, 81, 48, 54, 48, 112, 38, 60, 56, 108, 42, 96, 44, 96, 84, 72, 48, 156, 58, 96, 72, 112, 54, 132, 72, 144, 80, 90, 60, 192, 62, 96, 112, 166, 84

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000010, A000203, A010052, A046951, A076752 (Mobius transf.), A124315, A206369, A344442, A347090 (Dirichlet inverse).

Table[Sum[Boole[IntegerQ[(n/d)^(1/2)]] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] (EllipticTheta[3, 0, x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (2*p^(e + 3) - e*p^2 + e - If[OddQ[e], 3*p^2 - 1, 2*p^2 + 2*p - 2])/(2*(p - 1)*(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, May 25 2025 *)

A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d)); \\ Antti Karttunen, May 23 2021

G.f.: Sum_{k>=1} sigma(k) * (theta_3(x^k) - 1) / 2.
a(n) = Sum_{d|n} A076752(d).
a(n) = Sum_{d|n} A206369(n/d) * tau(d).
a(n) = Sum_{d|n} A010052(n/d) * sigma(d).
a(n) = Sum_{d|n} A124315(n/d) * phi(d).
a(n) = Sum_{d|n} A046951(n/d) * d.
a(p) = p + 1, where p is prime.
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1080. - Vaclav Kotesovec, Feb 26 2020
Multiplicative with a(p^e) = (2*p^(e+3) - e*p^2 + e - 3*p^2 + 1)/(2*(p-1)*(p^2-1)) if e is odd, and (2*p^(e+3) - e*p^2 + e - 2*p^2 - 2*p + 2)/(2*(p-1)*(p^2-1)) if e is even. - Amiram Eldar, May 25 2025

A347091 Sum of A332844 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 21, 16, 36, 0, 28, 0, 48, 48, 37, 0, 36, 0, 42, 64, 72, 0, 60, 36, 84, 48, 56, 0, 0, 0, 81, 96, 108, 96, 114, 0, 120, 112, 90, 0, 0, 0, 84, 72, 144, 0, 164, 64, 84, 144, 98, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 96, 166, 168, 0, 0, 126, 192, 0, 0, 258, 0, 228, 112, 140, 192, 0, 0, 246, 132

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A030059, A030229, A332844, A347090.

Cf. also A347093.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
v347090 = DirInverseCorrect(vector(up_to,n,A332844(n)));
A347090(n) = v347090[n];
A347091(n) = (A332844(n)+A347090(n));

a(n) = A332844(n) + A347090(n).
For n > 1, a(n) = -Sum_{d|n, 1A332844(d) * A347090(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A332844(A030229(n)).

cp's OEIS Frontend

A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

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