A374538 - OEIS

%I #7 Jul 12 2024 07:52:46

%S 1,5,10,1,26,50,50,65,1,130,122,10,170,250,260,1,290,5,362,26,500,610,

%T 530,650,1,850,730,50,842,1300,962,1025,1220,1450,1300,1,1370,1810,

%U 1700,1690,1682,2500,1850,122,26,2650,2210,10,1,5,2900,170,2810,3650,3172

%N a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

%C The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

%H Amiram Eldar, <a href="/A374538/b374538.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034676(A350389(n)).

%F a(n) >= 1 with equality if and only if n is a square (A000290).

%F a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).

%F Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.

%F Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).

%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

%t f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

%Y Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

%Y Cf. A002117, A013661, A183699.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jul 11 2024