A397238 - OEIS

A397238

Dirichlet g.f.: Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)).

1, 1, 1, 2, 1, 1, 1, 0, 3, 1, 1, 2, 1, 1, 1, 0, 1, 3, 1, 2, 1, 1, 1, 0, 5, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 6, 1, 1, 1, 0, 1, 1, 1, 2, 3, 1, 1, 0, 7, 5, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 1, 3, 1, 2, 1, 1, 1, 0, 1, 7, 3, 10

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OFFSET

1,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

FORMULA

Let f(s) = Product_{p prime} (p^s - 1) * (p^(2*s) - p) * (p^(2*s) + p^s + p) / p^(5*s).

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

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where

f(1) = Product_{p prime} (1 - 3/p^2 + 2/p^3) = A065473 = 0.28674742843447873410789271278983844643433184409705699564147785933665224...,

f'(1) = f(1) * Sum_{p prime} 9*log(p) / (p^2 + p - 2) = 1.20348507714464619427019986993136752898054316304997111175245238134639479...

and gamma is the Euler-Mascheroni constant A001620.

Multiplicative with a(p) = 1, a(p^2) = p, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Jul 21 2026

MATHEMATICA

f[p_, 1] := 1; f[p_, 2] := p; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 21 2026 *)

PROG

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X + p*X^2))[n], ", "))

CROSSREFS

Cf. A000189, A367990.

Cf. A069193, A363289, A384050.

Cf. A047994, A071773, A396336, A397136.

Cf. A065473.

Sequence in context: A319081 A336931 A363953 * A182662 A308778 A372472

Adjacent sequences: A397235 A397236 A397237 * A397240 A397241 A397242

KEYWORD

nonn,mult,easy

AUTHOR

Vaclav Kotesovec, Jul 21 2026

STATUS

approved