A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

0, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 4, 2, 4, 2, 3, 4, 3, 2, 4, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 4, 2, 3, 4, 2, 3, 3, 4, 2, 2, 2, 3, 4, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4

Offset: 1

Author

Paolo Xausa, Feb 17 2025

Keywords

• nonn
• easy

Comments

The prime factorization of 1 is the empty set, so a(1) = 0 by convention.

Examples

			a(16) = 2 because 12 = 2^3, the set of these bases and exponents is {2, 3} and its size is 2.
a(31500) = 5 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its size is 5.
		

Links

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

Crossrefs

Cf. A051674 (positions of ones), A381201, A381202, A381203, A381204, A381212.

Programs

Maple

a:= n-> nops({map(i-> i[], ifactors(n)[2])[]}):
seq(a(n), n=1..90);  # Alois P. Heinz, Feb 18 2025

Mathematica

A381205[n_] := If[n == 1, 0, Length[Union[Flatten[FactorInteger[n]]]]];
Array[A381205, 100]

PARI

a(n) = my(f=factor(n)); #setunion(Set(f[,1]), Set(f[,2])); \\ Michel Marcus, Feb 18 2025

Python

from sympy import factorint
def a(n): return len(set().union(*(set(pe) for pe in factorint(n).items())))
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Feb 18 2025