A050252 Number of digits in the prime factorization of n (counting terms of the form p^1 as p).

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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• base

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Factorization.

Crossrefs

Cf. A046758, A073048, A055642, A020639, A027748, A124010, A110475.

Cf. also A377369 (analog for base 2).

Programs

Haskell

a050252 1 = 1
a050252 n = sum $ map a055642 $
            (a027748_row n) ++ (filter (> 1) $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2013, Jun 21 2011

Mathematica

nd[n_]:=Total@IntegerLength@Select[Flatten@FactorInteger[n],#>1&];Table[If[n==1,1,nd[n]],{n,102}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

Python

from sympy import factorint
def a(n): return 1 if n == 1 else sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items())
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 27 2024

Formula

a(A192010(n)) = n and a(m) != n for m < A192010(n);

a(A046759(n)) < A055642(A046759(n)); a(A046758(n)) = A055642(A046758(n)); a(A046760(n)) > A055642(A046760(n)). [Reinhard Zumkeller, Jun 21 2011]