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

%I A381205 #18 Feb 22 2025 09:57:14
%S A381205 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,
%T A381205 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,
%U A381205 3,4,2,2,2,3,4,3,3,4,2,4,2,3,2,4,3,3,3,4,2,4
%N A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.
%C A381205 The prime factorization of 1 is the empty set, so a(1) = 0 by convention.
%H A381205 Paolo Xausa, <a href="/A381205/b381205.txt">Table of n, a(n) for n = 1..10000</a>
%e A381205 a(16) = 2 because 12 = 2^3, the set of these bases and exponents is {2, 3} and its size is 2.
%e A381205 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.
%p A381205 a:= n-> nops({map(i-> i[], ifactors(n)[2])[]}):
%p A381205 seq(a(n), n=1..90);  # _Alois P. Heinz_, Feb 18 2025
%t A381205 A381205[n_] := If[n == 1, 0, Length[Union[Flatten[FactorInteger[n]]]]];
%t A381205 Array[A381205, 100]
%o A381205 (PARI) a(n) = my(f=factor(n)); #setunion(Set(f[,1]), Set(f[,2])); \\ _Michel Marcus_, Feb 18 2025
%o A381205 (Python)
%o A381205 from sympy import factorint
%o A381205 def a(n): return len(set().union(*(set(pe) for pe in factorint(n).items())))
%o A381205 print([a(n) for n in range(1, 91)]) # _Michael S. Branicky_, Feb 18 2025
%Y A381205 Cf. A051674 (positions of ones), A381201, A381202, A381203, A381204, A381212.
%K A381205 nonn,easy
%O A381205 1,2
%A A381205 _Paolo Xausa_, Feb 17 2025