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A367932
a(n) is the smallest number k such that k*n is an exponentially evil number (A262675).
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
COMMENTS
First differs from A365298 at n = 64.
LINKS
FORMULA
Multiplicative with a(p^e) = p^s(e), s(e) = min{k >= e, k is evil} - e.
a(n) = A367934(n)/n.
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} f(1/p) = 0.623746285..., where f(x) = (1-x) * (1 + Sum_{k>=1} (x^(3*k-s(k))), and s(k) is defined above.
MATHEMATICA
f[p_, e_] := Module[{k = e}, While[! EvenQ[DigitCount[k, 2 , 1]], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(e) = {my(k = e); while(hammingweight(k)%2, k++); k - e; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }
CROSSREFS
Similar sequences: A365298, A365685, A367931.
Sequence in context: A358786 A360541 A365298 * A048798 A007914 A048758
KEYWORD
nonn,easy,mult,base
AUTHOR
Amiram Eldar, Dec 05 2023
STATUS
approved