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
KEYWORD
nonn,easy,mult
AUTHOR
Antti Karttunen, Nov 16 2019
STATUS
approved