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

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n.

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)

Example

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.

Mathematica

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

Prog

(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

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.

Sequence in context: A366078 A344652 A366074 * A144095 A362719 A076092

Adjacent sequences: A293436 A293437 A293438 * A293440 A293441 A293442

Keyword

nonn,easy

Author

Antti Karttunen, Nov 01 2017

Status

approved