A358347 - OEIS

%I #20 Sep 09 2023 06:49:35

%S 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,

%T 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,

%U 65,1,1,1,5,1,1,1,10,1,1,26,5,1,1,1,17,82,1

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

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

%H Amiram Eldar, <a href="/A358347/b358347.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

%F Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - _Amiram Eldar_, Jan 29 2023

%F a(n) = A034448(A350388(n)). - _Amiram Eldar_, Sep 09 2023

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

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

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

%Y Similar sequences: A033634, A034448, A035316, A358346.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Nov 11 2022