A377369 a(n) = total number of bits in the binary representation of the prime factorization of n (including exponents > 1).

0, 2, 2, 4, 3, 4, 3, 4, 4, 5, 4, 6, 4, 5, 5, 5, 5, 6, 5, 7, 5, 6, 5, 6, 5, 6, 4, 7, 5, 7, 5, 5, 6, 7, 6, 8, 6, 7, 6, 7, 6, 7, 6, 8, 7, 7, 6, 7, 5, 7, 7, 8, 6, 6, 7, 7, 7, 7, 6, 9, 6, 7, 7, 5, 7, 8, 7, 9, 7, 8, 7, 8, 7, 8, 7, 9, 7, 8, 7, 8, 5, 8, 7, 9, 8, 8, 7, 8, 7, 9

Offset

1,2

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

a(10) = 5 because 10 = 2*5 = 10_2*101_2 (5 total bits).
a(18) = 6 because 18 = 2*3^2 = 10_2*11_2^10_2 (6 total bits).

Mathematica

A377369[n_] := Total[BitLength[Select[Flatten[FactorInteger[n]], # > 1 &]]];
Array[A377369, 100]

Prog

(Python)
from sympy import factorint
def a(n): return sum(len(bin(p)[2:])+(len(bin(e)[2:]) if e>1 else 0) for p, e in factorint(n).items())
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Dec 27 2024
(PARI) a(n)={my(f=factor(n)); sum(i=1, #f~, 1 + logint(f[i, 1], 2) + (f[i, 2]>1) + logint(f[i, 2], 2))} \\ Andrew Howroyd, Dec 29 2024

Crossrefs

Cf. A050252 (analogous for base 10).

Cf. A070939.

Sequence in context: A338756 A078317 A105016 * A074747 A128248 A347659

Adjacent sequences: A377366 A377367 A377368 * A377370 A377371 A377372

Keyword

nonn,base

Author

Paolo Xausa, Dec 27 2024

Status

approved