A377369 - OEIS

%I #23 Jan 01 2025 18:13:33

%S 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,

%T 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,

%U 7,8,7,8,7,8,7,9,7,8,7,8,5,8,7,9,8,8,7,8,7,9

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

%H Paolo Xausa, <a href="/A377369/b377369.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 5 because 10 = 2*5 = 10_2*101_2 (5 total bits).

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

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

%t Array[A377369, 100]

%o (Python)

%o from sympy import factorint

%o 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())

%o print([a(n) for n in range(1, 91)]) # _Michael S. Branicky_, Dec 27 2024

%o (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

%Y Cf. A050252 (analogous for base 10).

%Y Cf. A070939.

%K nonn,base

%O 1,2

%A _Paolo Xausa_, Dec 27 2024