A350388 a(n) is the largest unitary divisor of n that is a square.

1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4

Offset: 1

Author

Amiram Eldar, Dec 28 2021

Keywords

• nonn
• easy
• mult

Comments

First differs from A056623 at n = 32.

Links

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

Crossrefs

Cf. A000290, A001222, A008833, A056623, A071974, A076479, A191414, A268335, A335275, A350386, A350389.

Programs

Mathematica

f[p_, e_] := If[EvenQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, f[i,1]^f[i,2]));} \\ Amiram Eldar, Oct 01 2023

Formula

Multiplicative with a(p^e) = p^e if e is even and 1 otherwise.

a(n) = n/A350389(n).

a(n) = A071974(n)^2.

a(n) = A008833(n) if and only if n is in A335275.

A001222(a(n)) = A350386(n).

a(n) = 1 if and only if n is an exponentially odd number (A268335).

a(n) = n if and only if n is a positive square (A000290 \ {0}).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1/3) * Product_{p prime} (1 + sqrt(p)/(1 + p + p^2)) = 0.59317173657411718128... [updated Oct 16 2022]

Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-2)). - Amiram Eldar, Oct 01 2023

Sum_{d|n, gcd(d, n/d) == 1} A076479(d) * a(n/d) = A191414(sqrt(n)) if n is a square, and 0 otherwise. - Amiram Eldar, Jun 01 2025