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%I #9 Jan 12 2024 22:52:12
%S 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0
%N The number of exponential divisors of n that are squares (A000290).
%H Amiram Eldar, <a href="/A368980/b368980.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A049419(n^2).
%F Multiplicative with a(p^e) = A183063(e), or equivalently, a(p^e) = 0 if e is odd, and A000005(e/2) if e is even.
%F a(n) >= 0, with equality if and only if n is not a square number (A000037).
%F a(n) <= A049419(n), with equality if and only if n = 1.
%F Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1.602317... (A327837).
%t f[p_, e_] := If[OddQ[e], 0, DivisorSigma[0, e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = vecprod(apply(x -> if(x%2, 0, numdiv(x/2)), factor(n)[, 2]));
%Y Cf. A000005, A000037, A000290, A049419, A183063, A322791, A327837, A368979.
%Y Similar sequences: A046951, A056624, A358260, A368978.
%K nonn,easy,mult
%O 1,16
%A _Amiram Eldar_, Jan 11 2024
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