A367990 Sum of the squarefree divisors of the largest unitary divisor of n that is a square.

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 07 2023

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000203, A048250, A268335, A336643, A350388, A367991.

Cf. A307868.

Programs

Mathematica

f[p_, e_] := If[EvenQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(!(f[i,2]%2), f[i,1]+1, 1));}

Formula

Multiplicative with a(p^e) = p + 1 if e is even and 1 otherwise.

a(n) = A048250(A350388(n)).

a(n) = A000203(A336643(n)).

a(n) = A048250(n)/A367991(n).

a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).

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

From Vaclav Kotesovec, Apr 20 2025: (Start)

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

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

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

f(1) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321167...

f'(1) = f(1) * Sum_{p prime} (7*p + 5) * log(p) / ((p-1)*(p+1)*(p+2)) = f(1) * 3.0570993566532132522378281945383016697995408795919384628849894110222383828...

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