A367513 - OEIS

%I #11 Nov 23 2023 14:34:00

%S 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,

%T 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,

%U 1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%H Amiram Eldar, <a href="/A367513/b367513.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

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

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, 1, f[i, 1]^f[i, 2]));}

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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

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

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

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Nov 21 2023