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
Examples
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).
Links
- Paolo Xausa, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
A377369[n_] := Total[BitLength[Select[Flatten[FactorInteger[n]], # > 1 &]]]; Array[A377369, 100]
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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 -
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