A056622 a(n) = A000188(n)/A055229(n).

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 1

Offset: 1

Author

Labos Elemer, Aug 08 2000

Keywords

• nonn
• easy
• mult

Comments

Previous name: "Square root of largest unitary square divisor of n." The previous name was incorrect for numbers that have an odd exponent in their prime factorization that is larger than 3. For the correct square root of largest unitary square divisor of n see A071974. - Amiram Eldar, Jul 26 2024

Multiplicative because quotient of two multiplicative sequences. - Christian G. Bower, May 16 2005

Examples

			For n = 125: A000188(125) = 5, A055229(125) = 5, so a(125) = 1.
For n = 360: A000188(360) = 6, A055229(360) = 2, so a(360) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000188, A055229, A034444, A056623, A071974.

Programs

Mathematica

f[p_, e_] := If[EvenQ[e], p^(e/2), If[e == 1, 1, p^((e - 3)/2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

PARI

A000188(n) = core(n, 1)[2]; \\ Michel Marcus, Feb 27 2013
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A056622(n) = (A000188(n)/A055229(n)); \\ Antti Karttunen, Nov 19 2017

Formula

Multiplicative with a(p^e) = p^(e/2) if e even, a(p) = 1, and a(p^e) = p^((e-3)/2) for odd e > 1. - Amiram Eldar, Sep 14 2020

Dirichlet g.f.: zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1) + 1/p^(3*s)). - Amiram Eldar, Dec 18 2023

a(n) = sqrt(A056623(n)). - Amiram Eldar, Jul 26 2024

From Vaclav Kotesovec, Jan 27 2025: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where

f(1) = Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 0.490798286634728225909154323920711804307234495196201399106047774...,

f'(1) = f(1) * Sum_{p prime} (5*p^2 - 7*p + 4) * log(p) / (p^4 - 2*p^2 + 2*p - 1) = f(1) * 1.94788222046256567576552118452630646598176999674201755783...

and gamma is the Euler-Mascheroni constant A001620. (End)

Extensions

Name replaced with a formula by Amiram Eldar, Jul 26 2024