A367699 The smallest divisor d of n such that n/d is an exponentially evil number (A262675).

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 6, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Amiram Eldar, Nov 27 2023

Keywords

• nonn
• easy
• mult
• base

Comments

First differs from A050985 at n = 32.

Links

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

Crossrefs

Cf. A050985, A262675, A366906, A367698.

Programs

Mathematica

maxEvil[e_] := Module[{k = e}, While[OddQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^(e - maxEvil[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = {my(k = n); while(hammingweight(k)%2, k--); n-k; }
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Formula

a(n) = n/A366906(n).

Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k evil}).

a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.84485885044273919581..., where f(x) = (1-x)*(1+Sum_{k>=1} x^(k+s(k))), s(k) is defined above for k >= 1, and s(0) = 0.