A367169 a(n) is the sum of the exponents in the prime factorization of n that are powers of 2.

0, 1, 1, 2, 1, 2, 1, 0, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 1, 2, 2, 0, 3, 1, 3, 1, 0, 2, 2, 2, 4, 1, 2, 2, 1, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A001222(A367168(n)).

Additive with a(p^e) = A048298(e).

a(n) <= A001222(n), with equality if and only if n is in A138302.

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} 2^k * (P(2^k) - P(2^k+1)) = 0.28425245481079272416..., where P(s) is the prime zeta function.

Mathematica

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), f[i, 2], 0)); }
(Python)
from sympy import factorint
def A367169(n): return sum(e for e in factorint(n).values() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Crossrefs

Cf. A001222, A048298, A077761, A138302, A367168, A367170, A367171.

Similar sequences: A350386, A350387.

Sequence in context: A105700 A378534 A366077 * A236831 A030205 A159817

Adjacent sequences: A367166 A367167 A367168 * A367170 A367171 A367172

Keyword

nonn,easy

Author

Amiram Eldar, Nov 07 2023

Status

approved