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

Author

Amiram Eldar, Jan 11 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

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

Similar sequences: A046951, A056624, A358260, A368978.

Programs

Mathematica

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

PARI

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

Formula

a(n^2) = A049419(n). [corrected by Ridouane Oudra, Nov 19 2024]

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).