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A365298
a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).
4
1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 2, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 18, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364
OFFSET
1,2
LINKS
FORMULA
Multiplicative with a(p) = p^2, a(p^e) = p if e is even, and a(p^e) = 1 is e is odd and > 1.
a(n) = A356192(n)/n.
a(n) = 1 if and only if n is in A335988.
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(3*s-2) - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (2*Pi^4/315) * Product_{p prime} (1 - p^2 - p^3 + p^4 + p^8 + p^9)/(p^8*(p+1)) = 0.207915752545... .
MATHEMATICA
f[p_, e_] := p^If[OddQ[e], Max[e, 3] - e, 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]^if(f[i, 2]%2, max(f[i, 2], 3) - f[i, 2], 1))};
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Aug 31 2023
STATUS
approved