A366906 The largest exponentially evil divisor of n.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Oct 27 2023

Keywords

• nonn
• easy
• mult

Comments

The largest divisor of n that is an exponentially evil number (A262675).

The number of exponentially evil divisors of n is A366902(n) and their sum is A366904(n).

Links

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

Crossrefs

Cf. A004709, A262675, A366902, A366904.

Similar sequences: A353897, A365683, A366905.

Programs

Mathematica

maxEvil[e_] := Module[{k = e}, While[OddQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^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--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Formula

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

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

a(n) >= 1, with equality if and only if n is a cubefree number (A004709).