A293439 - OEIS

%I #23 Nov 23 2023 12:14:05

%S 0,1,1,1,1,2,1,0,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,1,1,2,0,2,1,3,1,0,2,2,

%T 2,2,1,2,2,1,1,3,1,2,2,2,1,2,1,2,2,2,1,1,2,1,2,2,1,3,1,2,2,0,2,3,1,2,

%U 2,3,1,1,1,2,2,2,2,3,1,2,1,2,1,3,2,2,2,1,1,3,2,2,2,2,2,1,1,2,2,2,1,3,1,1,3

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

%H Antti Karttunen, <a href="/A293439/b293439.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F Additive with a(p^e) = A010060(e).

%F a(n) = A007814(A293443(n)).

%F From _Amiram Eldar_, Sep 28 2023: (Start)

%F a(n) >= 0, with equality if and only if n is an exponentially evil number (A262675).

%F a(n) <= A001221(n), with equality if and only if n is an exponentially odious number (A270428).

%F Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = -0.12689613844142998028..., where f(x) = 1/2 - x - ((1-x)/2) * Product_{k>=0} (1-x^(2^k)). (End)

%e For n = 2 = 2^1, the only exponent 1 is odious (that is, has an odd Hamming weight and thus is included in A000069), so a(2) = 1.

%e For n = 24 = 2^3 * 3^1, the exponent 3 (with binary representation "11") is evil (has an even Hamming weight and thus is included in A001969), while the other exponent 1 is odious, so a(24) = 1.

%t a[n_] := Total@ ThueMorse[FactorInteger[n][[;; , 2]]]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, May 18 2023 *)

%o (PARI) A293439(n) = vecsum(apply(e -> (hammingweight(e)%2), factorint(n)[, 2]));

%o (Python)

%o from sympy import factorint

%o def A293439(n): return sum(1 for e in factorint(n).values() if e.bit_count()&1) # _Chai Wah Wu_, Nov 23 2023

%Y Cf. A000069, A010060, A077761, A262675, A293443.

%Y Cf. A270428 (numbers such that a(n) = A001221(n)).

%Y Differs from A144095 for the first time at n=24.

%K nonn,easy

%O 1,6

%A _Antti Karttunen_, Nov 01 2017