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

1,4

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A334039 A076933 A071974 * A375568 A331738 A372328

Adjacent sequences: A056619 A056620 A056621 * A056623 A056624 A056625

Keyword

nonn,easy,mult

Author

Labos Elemer, Aug 08 2000

Extensions

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

Status

approved