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

1,2

Comments

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

Links

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

Example

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.

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]

Prog

(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

Crossrefs

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

Sequence in context: A216817 A263765 A335424 * A270073 A027348 A238325

Adjacent sequences: A381202 A381203 A381204 * A381206 A381207 A381208

Keyword

nonn,easy,new

Author

Paolo Xausa, Feb 17 2025

Status

approved