A367932 a(n) is the smallest number k such that k*n is an exponentially evil number (A262675).

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

Author

Amiram Eldar, Dec 05 2023

Keywords

• nonn
• easy
• mult
• base

Comments

First differs from A365298 at n = 64.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001969, A262675, A367934.

Similar sequences: A365298, A365685, A367931.

Programs

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]

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])); }

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.