A368980 The number of exponential divisors of n that are squares (A000290).

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset

1,16

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A049419(n^2).

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.

a(n) >= 0, with equality if and only if n is not a square number (A000037).

a(n) <= A049419(n), with equality if and only if n = 1.

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1.602317... (A327837).

Mathematica

f[p_, e_] := If[OddQ[e], 0, DivisorSigma[0, e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> if(x%2, 0, numdiv(x/2)), factor(n)[, 2]));

Crossrefs

Cf. A000005, A000037, A000290, A049419, A183063, A322791, A327837, A368979.

Similar sequences: A046951, A056624, A358260, A368978.

Sequence in context: A138045 A072507 A004530 * A355549 A347438 A245196

Adjacent sequences: A368977 A368978 A368979 * A368981 A368982 A368983

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 11 2024

Status

approved