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A370296
Inverse Moebius transform of A322327.
0
1, 3, 3, 7, 3, 9, 3, 13, 7, 9, 3, 21, 3, 9, 9, 21, 3, 21, 3, 21, 9, 9, 3, 39, 7, 9, 13, 21, 3, 27, 3, 31, 9, 9, 9, 49, 3, 9, 9, 39, 3, 27, 3, 21, 21, 9, 3, 63, 7, 21, 9, 21, 3, 39, 9, 39, 9, 9, 3, 63, 3, 9, 21, 43, 9, 27, 3, 21, 9, 27, 3, 91, 3, 9, 21, 21, 9, 27, 3, 63, 21, 9, 3, 63
OFFSET
1,2
FORMULA
Multiplicative with a(p^e) = 1 + e + e^2 for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).
Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -3 and b(p^e) = 2 * (-1)^((e+1)*(e+2)/2) for prime p and e > 1.
Dirichlet convolution of A000005 and A323308.
MATHEMATICA
f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)
PROG
(PARI) a(n) = factorback(apply(e->1+e+e^2, factor(n)[, 2]))
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Werner Schulte, Feb 14 2024
STATUS
approved