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

1,2

Comments

First differs from A365298 at n = 64.

Links

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

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

Cf. A001969, A262675, A367934.

Similar sequences: A365298, A365685, A367931.

Sequence in context: A358786 A360541 A365298 * A048798 A007914 A048758

Adjacent sequences: A367929 A367930 A367931 * A367933 A367934 A367935

Keyword

nonn,easy,mult,base

Author

Amiram Eldar, Dec 05 2023

Status

approved