A056622 Square root of largest unitary square divisor of 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

Unitary analog of A000188. These numbers are neither unitary nor necessarily square divisors.

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

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

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

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

Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)
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.

Sequence in context: A334039 A076933 A071974 * A331738 A306333 A237983

Adjacent sequences: A056619 A056620 A056621 * A056623 A056624 A056625

Keyword

nonn,easy,mult

Author

Labos Elemer, Aug 08 2000

Status

approved