A367513 The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 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, 1, 1, 1, 1, 1, 1, 1

Offset

1,8

Links

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

Formula

Multiplicative with a(p^e) = p^(e*A010059(e)) = p^A102391(e).

a(n) = n/A367514(n).

A001221(a(n)) = A367512(n).

A034444(a(n)) = A367516(n).

a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).

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

Mathematica

f[p_, e_] := p^(e * (1 - ThueMorse[e])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, 1, f[i, 1]^f[i, 2])); }
(Python)
from math import prod
from sympy import factorint
def A367513(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Crossrefs

Cf. A001969, A010059, A034444, A102391, A262675, A270428, A366902, A366904, A367512, A367516.

Similar sequences: A350388, A350389, A366906, A367168, A367514.

Sequence in context: A366126 A056191 A368167 * A368540 A103760 A268355

Adjacent sequences: A367510 A367511 A367512 * A367514 A367515 A367516

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 21 2023

Status

approved