A367512 - OEIS

%I #12 Nov 23 2023 15:16:55

%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,

%T 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0

%N Number of evil exponents in the prime factorization of n.

%H Amiram Eldar, <a href="/A367512/b367512.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = A010059(e).

%F a(n) = A001221(n) - A293439(n).

%F a(n) = A001221(A367513(n)).

%F a(n) = log_2(A367516(n)).

%F a(n) >= 0, with equality if and only if n is an exponentially odious number (A270428).

%F a(n) <= A001221(n), with equality if and only if n is an exponentially evil number (A262675).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.12689613844142998028..., where f(x) = x - 1/2 + ((1-x)/2) * Product_{k>=0} (1-x^(2^k)).

%t f[p_, e_] := 1 - ThueMorse[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecsum(apply(e -> !(hammingweight(e)%2), factor(n)[, 2]));

%o (Python)

%o from sympy import factorint

%o def A367512(n): return sum(1 for e in factorint(n).values() if e.bit_count()&1^1) # _Chai Wah Wu_, Nov 23 2023

%Y Cf. A001221, A010059, A262675, A270428, A293439, A367513, A367516.

%K nonn,easy

%O 1

%A _Amiram Eldar_, Nov 21 2023