A090140 Number of groups of order 7^n.

1, 1, 2, 5, 15, 83, 860, 113147

Offset

0,3

References

G. Bagnera, La composizione dei Gruppi finiti il cui grado e la quinta potenza di un numero primo, Ann. Mat. Pura Appl. (3), 1 (1898), 137-228.

Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, International Journal of Algebra and Computation, Vol. 12, No 5 (2002), 623-644.

W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955.

M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra, 278 (2004), 383-401.

E. A. O'Brien and M. R. Vaughan-Lee, The groups of order p^7 for odd prime p, J. Algebra 292, 243-258, 2005.

Links

Table of n, a(n) for n=0..7.

Formula

For a prime p >= 5, the number of groups of order p^n begins 1, 1, 2, 5, 15, 61 + 2*p + 2*gcd (p - 1, 3) + gcd (p - 1, 4), 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5), ...

Crossrefs

Cf. A000001, A000679, A090091, A090130.

Sequence in context: A268122 A133645 A090130 * A032267 A191477 A196196

Adjacent sequences: A090137 A090138 A090139 * A090141 A090142 A090143

Keyword

nonn

Author

Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004

Extensions

Corrected and extended by David Radcliffe, Feb 24 2010

Updated reference for p^7 Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Mar 06 2010

Status

approved