Eamonn O'Brien - cp's OEIS frontend

User: Eamonn O'Brien

Eamonn O'Brien's wiki page.

Eamonn O'Brien has authored 3 sequences.

A090140 Number of groups of order 7^n.

Original entry on oeis.org

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

Offset: 0

Eamonn O'Brien, Jan 22 2004

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.

Cf. A000001, A000679, A090091, A090130.

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), ...

Corrected and extended by David Radcliffe, Feb 24 2010

Updated reference for p^7 Eamonn O'Brien, Mar 06 2010

A090091 Number of groups of order 3^n.

Original entry on oeis.org

1, 1, 2, 5, 15, 67, 504, 9310, 1396077, 5937876645

Offset: 0

Eamonn O'Brien, Jan 22 2004

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 67*x^5 + 504*x^6 + 9310*x^7 + ...

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.
Marcus du Sautoy, Symmetry: A Journey into the Patterns of Nature, HarperCollins, 2008, p. 96.

David Burrell, The number of p-groups of order 19,683 and new lists of p-groups, Communications in Algebra, Vol. 51 - Issue 6 (2023), 2673-2679.
Heiko Dietrich, Computational aspects of finite p-groups, 2016.
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
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. [_David Radcliffe_, Feb 24 2010]
Michael Vaughan-Lee, Graham Higman’s PORC Conjecture, Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 114 (2012), 89-16.
Michael Vaughan-Lee, Groups of order p^8 and exponent p, International Journal of Group Theory Vol. 4 (2015), 25-42.
Brett Edward Witty, Enumeration of groups of prime-power order, PhD thesis, 2006.

Cf. A000001, A000679, A090130, A090140.

A090091 := List([0..7],n -> NumberSmallGroups(3^n)); # Muniru A Asiru, Oct 15 2017

with(GroupTheory): seq(NumGroups(3^n),n=0..8); # Muniru A Asiru, Oct 17 2018

a(n) = A000001(3^n).

a(7) from David Radcliffe, Feb 24 2010
a(8) from Muniru A Asiru, Oct 17 2018
a(9) from David Burrell, Sep 01 2023

A090130 Number of groups of order 5^n.

Original entry on oeis.org

1, 1, 2, 5, 15, 77, 684, 34297

Offset: 0

Eamonn O'Brien, Jan 22 2004

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. - _Eamonn O'Brien_, Mar 06 2010

Cf. A000001, A000679, A090091, A090140.

A090130 := List([0..7],n -> NumberSmallGroups(5^n)); # Muniru A Asiru, Oct 15 2017

Corrected and extended by David Radcliffe, Feb 24 2010

Corrected and extended by Eamonn O'Brien, Mar 06 2010

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User: Eamonn O'Brien

A090140 Number of groups of order 7^n.

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A090091 Number of groups of order 3^n.

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A090130 Number of groups of order 5^n.

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