A076558 a(n) = r * min(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n, a(1) = 0.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4

Offset: 1

Author

Joseph L. Pe, Nov 10 2002

Keywords

• easy
• nonn

Comments

Omega(n) >= a(n) for n >= 1, where Omega(n) = the number of prime factors of n, counting multiplicity.

Positions of records are A000079. - David A. Corneth, May 05 2020

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Carlos Rivera, Puzzle #201 The Arithmetic Function A(n), The Prime Puzzles and Problems Connection.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000079, A001221, A001222, A051904, A076526.

Programs

Mathematica

a[n_] := Module[{pf}, pf = Transpose[FactorInteger[n]]; Length[pf[[1]]]*Min[pf[[2]]]]; Table[a[i] - Boole[i == 1], {i, 100}]
(* Second program: *)
Table[Length[#] Min[#] - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

PARI

a(n) = if(n == 1, 0, my(e = factor(n)[, 2]); vecmin(e) * #e); \\ Amiram Eldar, Sep 08 2024

Python

from sympy import factorint
def a(n):
    f=factorint(n)
    l=[f[p] for p in f]
    return 0 if n==1 else len(l)*min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

Formula

a(n) = A001221(n) * A051904(n). - Antti Karttunen, Jul 12 2017

Extensions

a(1)=0 prepended by Antti Karttunen, Jul 12 2017