A066060 -id:A066060 - cp's OEIS frontend

A069739 Size of the key space for isomorphism verification of circulant graphs of order n.

1, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 14, 1, 2, 1, 2, 1, 1, 1, 5, 2, 1, 5, 2, 1, 1, 1, 42, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 14, 2, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 2, 1, 1, 2, 132, 1, 1, 1, 2, 1, 1, 1, 10, 1, 1, 2, 2, 1, 1, 1, 14, 14, 1, 1, 2, 1, 1, 1, 5, 1, 2

Offset: 1

Valery A. Liskovets, Apr 15 2002

Multiplicative with a(p^m) = Catalan(m) (A000108). Coincides with A066060 up to n=63 except for n=32.

Antti Karttunen, Table of n, a(n) for n = 1..10000
M. Muzychuk, A solution of the isomorphism problem for circulant graphs, Proc. London Math. Soc. (3) 88 (2004), no. 1, 1-41.
Index entries for sequences computed from exponents in factorization of n

Cf. A000108, A066060, A028234, A067029, A156552, A246596.

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A069739 := proc(n) local ifa; if n = 1 then 1; else ifa := ifactors(n)[2] ; mul( A000108(op(2,f)), f=ifa) ; end if; end proc:
seq(A069739(n),n=1..90) ; # R. J. Mathar, Feb 08 2011

Table[Times @@ Map[CatalanNumber, FactorInteger[n][[All, -1]]], {n, 90}] (* Michael De Vlieger, May 28 2017 *)

(define (A069739 n) (if (= 1 n) n (* (A000108 (A067029 n)) (A069739 (A028234 n))))) ;; Antti Karttunen, May 28 2017

a(n) = prod_p Catalan(m_p) where n=prod_p p^(m_p), p|n prime.
From Antti Karttunen, May 28-29 2017: (Start)
a(1) = 1; for n > 1, a(n) = A000108(A067029(n)) * a(A028234(n)).
a(n) = A246596(A156552(n)). (End)

A088806 Number of non-Abelian nilpotent groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 9, 10, 0, 0, 0, 0, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 22 2003

nonn

a(n) = 0 precisely when n is cubefree. - Eric M. Schmidt, Dec 19 2014

T. D. Noe, Table of n, a(n) for n = 1..2015

Cf. A066060, A000688.

a(n) = A066060(n) - A000688(n).

A384609 Possible values for the number of nilpotent groups of a finite order, ordered by size.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 15, 16, 20, 25, 28, 30, 32, 40, 50, 51, 56, 60, 64, 67, 70, 75, 77, 80, 83, 87, 97, 100, 101, 102, 107, 111, 112, 120, 125, 128, 131, 134, 140, 145, 149, 150, 154, 155, 159, 160, 166, 173, 174, 183, 193, 194, 196, 200, 202, 203, 204, 207

Offset: 1

Robin Jones, Jun 04 2025

nonn

A066060 sorted and duplicates removed.

List of all possible products of terms in A384607 (possibly with use of the same integer more than once).

			1 is in this sequence as there is exactly 1 nilpotent group of order 1.
2 is in this sequence as there are exactly 2 nilpotent groups of order 4.
4 is in this sequence as there are exactly 4 nilpotent groups of order 36.
3 is not in this sequence as there are never exactly 3 nilpotent groups of any given order.

Robin Jones, Table of n, a(n) for n = 1..135

Cf. A066060, A384607.

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A069739 Size of the key space for isomorphism verification of circulant graphs of order n.

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A088806 Number of non-Abelian nilpotent groups of order n.

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A384609 Possible values for the number of nilpotent groups of a finite order, ordered by size.

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