A358347 a(n) is the sum of the unitary divisors of n that are squares.

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1

Offset: 1

Author

Amiram Eldar, Nov 11 2022

Keywords

• nonn
• easy
• mult

Comments

The number of unitary divisors of n that are squares is A056624(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A056624, A077610, A078434, A247041, A268335, A350388, A351568.

Similar sequences: A033634, A034448, A035316, A358346.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], 1, p^e + 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, 1, f[i,1]^f[i,2] + 1));}

Formula

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

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

a(n) = A034448(n)/A358346(n).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)) = 0.6491241554... .

Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - Amiram Eldar, Jan 29 2023

a(n) = A034448(A350388(n)). - Amiram Eldar, Sep 09 2023