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A367513
The exponentially evil part of n: the largest unitary divisor of n that is an exponentially evil number (A262675).
5
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
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
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Nov 21 2023
STATUS
approved