A367988 - OEIS

%I #7 Dec 07 2023 10:25:39

%S 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,7,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,1,1,1,

%T 1,12,1,1,1,1,1,1,1,3,4,1,1,7,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,15,1,1,1,

%U 3,1,1,1,4,1,1,6,3,1,1,1,7,13,1,1,3,1,1

%N The sum of the divisors of the square root of the largest unitary divisor of n that is a square.

%H Amiram Eldar, <a href="/A367988/b367988.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.

%F a(n) = A000203(A071974(n)).

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

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

%t f[p_, e_] := If[EvenQ[e], (p^(e/2 + 1) - 1)/(p - 1), 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]/2 + 1) - 1)/(f[i,1] - 1)));}

%Y Cf. A000203, A071974, A268335, A351568, A367987.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Dec 07 2023