%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