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

Author

Antti Karttunen, Nov 16 2019

Keywords

• nonn
• easy
• mult

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)

Crossrefs

Cf. A000196, A062503, A337050.

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

Programs

Mathematica

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

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

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)