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A374537
a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.
2
1, 5, 10, 5, 26, 50, 50, 69, 10, 130, 122, 50, 170, 250, 260, 69, 290, 50, 362, 130, 500, 610, 530, 690, 26, 850, 739, 250, 842, 1300, 962, 1093, 1220, 1450, 1300, 50, 1370, 1810, 1700, 1794, 1682, 2500, 1850, 610, 260, 2650, 2210, 690, 50, 130, 2900, 850, 2810
OFFSET
1,2
COMMENTS
The number of divisors of n that are exponentially odd is A322483(n) and their sum is A033634(n).
LINKS
FORMULA
a(n) = A001157(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + p^2 * (p^(4*floor((e-1)/2)+4) - 1) / (p^4 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4)).
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) = A183699 * A065464 = 0.84677961058798544766... .
MATHEMATICA
f[p_, e_] := 1 + p^2 * (p^(4*Floor[(e-1)/2]+4) - 1) / (p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, 1 + p[i]^2 * (p[i]^(4*((e[i]-1)\2)+4) - 1) / (p[i]^4 - 1)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jul 11 2024
STATUS
approved