A329376 Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

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

Offset

1,4

Comments

Product of those distinct prime factors that occur exactly twice in the prime factorization of n, that is, whose exponent is 2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Formula

Multiplicative with a(p^e) = p when e = 2, otherwise a(p^e) = 1.

a(n) <= A000196(n).

From Amiram Eldar, Feb 11 2023: (Start)

a(n) <= sqrt(n), with equality if and only if n is in A062503.

a(n) = 1 if and only if n is in A337050. (End)

From Vaclav Kotesovec, May 31 2024: (Start)

Dirichlet g.f.: zeta(2*s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*s)).

Let f(s) = Product_{p prime} (1 - 1/p^(5*s-1) + 1/p^(5*s-2) + 1/p^(4*s-1) - 1/p^(4*s-2) - 1/p^(3*s-1) + 1/p^(3*s) - 1/p^(2*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 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,

f'(1) = f(1) * Sum_{p prime} (9*p^2 - 12*p + 5) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 3.78385641685861932254178374972226733621783278751462026270346293...

and gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

f[p_, e_] := If[e == 2, p, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 11 2023 *)

Prog

(PARI) A329376(n) = { my(f = factor(n)); prod(i=1, #f~, f[i, 1]^(2 == f[i, 2])); };
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X^2 + X + X^2/(-1 + 1/X)))[n], ", ")) \\ Vaclav Kotesovec, May 31 2024

Crossrefs

Cf. A000196, A062503, A337050.

Row 3 of array A106177, and the square roots of its row 9.

Sequence in context: A211005 A162154 A134505 * A336643 A334039 A076933

Adjacent sequences: A329373 A329374 A329375 * A329377 A329378 A329379

Keyword

nonn,easy,mult

Author

Antti Karttunen, Nov 16 2019

Status

approved