A293439 Number of odious exponents in the prime factorization of n.

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, 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, 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

Offset: 1

Author

Antti Karttunen, Nov 01 2017

Keywords

• nonn
• easy

Examples

			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.
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

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

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

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

Programs

Mathematica

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

PARI

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

Python

from sympy import factorint
def A293439(n): return sum(1 for e in factorint(n).values() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Formula

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

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

From Amiram Eldar, Sep 28 2023: (Start)

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

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

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)