A318306 Additive with a(p^e) = A002487(e).

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

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

Formula

a(n) = A007814(A318307(n)).

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) = 0.15790080909728804399..., where f(x) = -x + x * (1-x) * Product{k>=0} (1 + x^(2^k) + x^(2^(k + 1))). - Amiram Eldar, Feb 11 2024

Prog

(PARI)
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318306(n) = vecsum(apply(e -> A002487(e), factor(n)[, 2]));
(Python)
from functools import reduce
from sympy import factorint
def A318306(n): return sum(sum(reduce(lambda x, y:(x[0], x[0]+x[1]) if int(y) else (x[0]+x[1], x[1]), bin(e)[-1:2:-1], (1, 0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

Crossrefs

Cf. A002487, A077761, A318307.

Cf. also A046644.

Sequence in context: A058061 A371090 A064547 * A345935 A214715 A244145

Adjacent sequences: A318303 A318304 A318305 * A318307 A318308 A318309

Keyword

nonn

Author

Antti Karttunen, Aug 29 2018

Status

approved