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A367989
The sum of square divisors of the largest unitary divisor of n that is a square.
1
1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 21, 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, 21, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1
OFFSET
1,4
LINKS
FORMULA
a(n) = A035316(A350388(n)).
Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) if e is even and 1 otherwise.
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2)) = 0.69451968056653021193... .
MATHEMATICA
f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 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] + 2) - 1)/(f[i, 1]^2 - 1))); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 07 2023
STATUS
approved