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

Author

Amiram Eldar, Nov 21 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

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

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

Programs

Mathematica

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

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

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).