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

1,4

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

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

Mathematica

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

Prog

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

Crossrefs

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

Similar sequences: A033634, A034448, A035316, A358346.

Sequence in context: A028315 A074062 A094635 * A367989 A220054 A263152

Adjacent sequences: A358344 A358345 A358346 * A358348 A358349 A358350

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 11 2022

Status

approved