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A369428 The number of exponents in the prime factorization of n that are squares.

%I A369428 #13 Apr 26 2025 20:54:19
%S A369428 0,1,1,0,1,2,1,0,0,2,1,1,1,2,2,1,1,1,1,1,2,2,1,1,0,2,0,1,1,3,1,0,2,2,
%T A369428 2,0,1,2,2,1,1,3,1,1,1,2,1,2,0,1,2,1,1,1,2,1,2,2,1,2,1,2,1,0,2,3,1,1,
%U A369428 2,3,1,0,1,2,1,1,2,3,1,2,1,2,1,2,2,2,2
%N A369428 The number of exponents in the prime factorization of n that are squares.
%C A369428 First differs from A125070 at n = 64.
%H A369428 Amiram Eldar, <a href="/A369428/b369428.txt">Table of n, a(n) for n = 1..10000</a>
%F A369428 Additive with a(p^e) = A010052(e).
%F A369428 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = P(2) + Sum_{k>=2} (P(k^2+1) - P(k^2)) = 0.40999077396414387641..., and P(s) is the prime zeta function.
%p A369428 a:= n-> add(`if`(issqr(i[2]), 1, 0), i=ifactors(n)[2]):
%p A369428 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jan 23 2024
%t A369428 f[p_, e_] := If[IntegerQ[Sqrt[e]], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369428 (PARI) a(n) = vecsum(apply(x -> if(issquare(x), 1, 0), factor(n)[, 2]));
%Y A369428 Cf. A000290, A010052, A077761, A197680.
%Y A369428 Similar sequences: A125070, A293439, A366074, A367512, A369426, A369427.
%K A369428 nonn,easy
%O A369428 1,6
%A A369428 _Amiram Eldar_, Jan 23 2024