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A365334
The sum of exponentially odd divisors of the largest square dividing n.
2
1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 11, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 11, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 11, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 43, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 11, 31, 1, 1, 3, 1
OFFSET
1,4
COMMENTS
The number of these divisors is A365333(n).
LINKS
FORMULA
a(n) = A033634(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + (p^(e + 1 - (e mod 2)) - 1)/(p^2 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1)).
MATHEMATICA
f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - p)/(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~, (f[i, 1]^(f[i, 2] + 1 - f[i, 2]%2) - f[i, 1])/(f[i, 1]^2 - 1) + 1); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 01 2023
STATUS
approved