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Search: a090140 -id:a090140
Displaying 1-7 of 7 results found. page 1
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A090091 Number of groups of order 3^n. +10
10
1, 1, 2, 5, 15, 67, 504, 9310, 1396077, 5937876645 (list; graph; refs; listen; history; text; internal format)
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.
Marcus du Sautoy, Symmetry: A Journey into the Patterns of Nature, HarperCollins, 2008, p. 96.
LINKS
Table of n, a(n) for n=0..9.
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.
FORMULA
a(n) = A000001(3^n).
EXAMPLE
G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 67*x^5 + 504*x^6 + 9310*x^7 + ...
MAPLE
with(GroupTheory): seq(NumGroups(3^n), n=0..8); # Muniru A Asiru, Oct 17 2018
PROG
(GAP) A090091 := List([0..7], n -> NumberSmallGroups(3^n)); # Muniru A Asiru, Oct 15 2017
CROSSREFS
Cf. A000001, A000679, A090130, A090140.
KEYWORD
nonn,hard,more
AUTHOR
Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004
EXTENSIONS
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
STATUS
approved
A090130 Number of groups of order 5^n. +10
8
1, 1, 2, 5, 15, 77, 684, 34297 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
Table of n, a(n) for n=0..7.
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]
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), ...
PROG
(GAP) A090130 := List([0..7], n -> NumberSmallGroups(5^n)); # Muniru A Asiru, Oct 15 2017
CROSSREFS
Cf. A000001, A000679, A090091, A090140.
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
Corrected and extended by Eamonn O'Brien, Mar 06 2010
STATUS
approved
A232106 Number of groups of order prime(n)^6. +10
7
267, 504, 684, 860, 1192, 1476, 1944, 2264, 2876, 4068, 4540, 6012, 7064, 7664, 8852, 10908, 13136, 14012, 16520, 18292, 19296, 22244, 24296, 27648, 32472, 34964, 36284, 38912, 40356, 43128, 53780, 56992, 62064, 63824, 72828, 74740, 80532, 86504, 90572, 96948 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Isomorphism types of groups and nilpotent Lie rings with order prime(n)^6.
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to groups
FORMULA
For a prime p > 3, the number of groups of order p^6 is 3p^2 + 39p + 344 + 24 gcd(p - 1, 3) + 11 gcd(p - 1, 4) + 2 gcd(p - 1, 5).
MAPLE
a:= n-> `if`(n<3, [267, 504][n], (c-> 386 +(45 +3*c)*c+
24*igcd(c, 3) +11*igcd(c, 4) +2*igcd(c, 5))(ithprime(n)-1)):
seq(a(n), n=1..40); # Alois P. Heinz, Nov 17 2017
MATHEMATICA
Table[FiniteGroupCount[Prime[n]^6], {n, 40}] (* Michael De Vlieger, Apr 12 2016 *)
PROG
(Sage) def A232106(n) : p = nth_prime(n); return 267 if p==2 else 504 if p==3 else 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5)
(PARI) a(n) = if(n==1, 267, if (n==2, 504, my(p=prime(n)); 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5))); \\ Altug Alkan, Apr 12 2016
(GAP) A232106 := Concatenation([267, 504], List(Filtered([5..10^5], IsPrime), p -> 3 * p^2 + 39 * p + 344 + 24 * Gcd(p-1, 3) + 11 * Gcd(p-1, 4) + 2 * Gcd(p-1, 5))); # Muniru A Asiru, Nov 16 2017
CROSSREFS
Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232107.
Cf. A030516.
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Nov 21 2013
STATUS
approved
A232105 Number of groups of order prime(n)^5. +10
4
51, 67, 77, 83, 87, 97, 101, 107, 111, 125, 131, 145, 149, 155, 159, 173, 183, 193, 203, 207, 217, 227, 231, 245, 265, 269, 275, 279, 289, 293, 323, 327, 341, 347, 365, 371, 385, 395, 399, 413, 423, 433, 447, 457, 461, 467, 491, 515, 519, 529, 533, 543, 553 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 1..1000
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.
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.
Index entries for sequences related to groups
FORMULA
For a prime p > 3, the number of groups of order p^5 is 61 + 2p + 2 gcd(p - 1, 3) + gcd(p - 1, 4).
PROG
(Sage) def A232105(n) : p = nth_prime(n); return 51 if p==2 else 67 if p==3 else 61 + 2*p + 2*gcd(p - 1, 3) + gcd(p - 1, 4)
(GAP) A232105 := Concatenation([51, 67], List(Filtered([5..10^5], IsPrime), p -> 61 + 2 * p + 2 * Gcd(p-1, 3) + Gcd(p-1, 4))); # Muniru A Asiru, Nov 16 2017
CROSSREFS
Cf. A000001, A000679, A090091, A090130, A090140, A232106, A232107.
Cf. A050997.
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Nov 21 2013
STATUS
approved
A232107 Number of groups of order prime(n)^7. +10
4
2328, 9310, 34297, 113147, 750735, 1600573, 5546909, 9380741, 23316851, 71271069, 98488755, 233043067, 384847485, 485930975, 751588475, 1356370173, 2299880351, 2710679045, 4306310927, 5734323819, 6578172579, 9721485395, 12413061671, 17537591045, 26866372821 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to groups
FORMULA
For a prime p > 5, the number of groups of order p^7 is 3p^5 + 12p^4 + 44p^3 + 170p^2 + 707p + 2455 + (4p^2 + 44p + 291)gcd(p - 1, 3) + (p^2 + 19p + 135)gcd(p - 1, 4) + (3p + 31)gcd(p - 1, 5) + 4 gcd(p - 1, 7) + 5 gcd(p - 1, 8) + gcd(p - 1, 9).
MAPLE
a:= n-> `if`(n<4, [2328, 9310, 34297][n], (c-> 3391 +(1242+
(404 +(122 +(27 +3*c)*c)*c)*c)*c +(339 +(52 +4*c)*c)*igcd(c, 3)+
(155 +(21 +c)*c)*igcd(c, 4) +(34 +3*c)*igcd(c, 5) +4*igcd(c, 7)+
5*igcd(c, 8) +igcd(c, 9))(ithprime(n)-1)):
seq(a(n), n=1..25); # Alois P. Heinz, Nov 17 2017
PROG
(Sage) def A232107(n) : p = nth_prime(n); return 2328 if p==2 else 9310 if p==3 else 34297 if p==5 else 3*p^5 + 12*p^4 + 44*p^3 + 170*p^2 + 707*p + 2455 + (4*p^2 + 44*p + 291)*gcd(p - 1, 3) + (p^2 + 19*p + 135)*gcd(p - 1, 4) + (3*p + 31)*gcd(p - 1, 5) + 4*gcd(p - 1, 7) + 5*gcd(p - 1, 8) + gcd(p - 1, 9)
(GAP) A232107 := Concatenation([2328, 9310, 34297], List(Filtered([7..10^5], IsPrime), p -> 3 * p^5 + 12 * p^4 + 44 * p^3 + 170 * p^2 + 707 * p + 2455 + (4 * p^2 + 44 * p + 291) * Gcd(p-1, 3) + (p^2 + 19 * p + 135) * Gcd(p-1, 4) + (3 * p + 31) * Gcd(p-1, 5) + 4 * Gcd(p-1, 7) + 5 * Gcd(p-1, 8) + Gcd(p-1, 9))); # Muniru A Asiru, Nov 16 2017
CROSSREFS
Cf. A000001, A000679, A090091, A090130, A090140, A232105, A232106.
Cf. A092759.
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Nov 21 2013
STATUS
approved
A252787 Number of exponent-7 class 2 groups of order 7^n. +10
3
0, 0, 1, 3, 7, 28, 131, 26465, 2832610615, 18602165186505644, 1577146952504975321253720851 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..10.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
E. A. O'Brien, Bibliography on the determination of finite groups.
CROSSREFS
Cf. A090140, A252784, A252785, A252786.
KEYWORD
nonn,hard,more
AUTHOR
Eric M. Schmidt, Dec 21 2014
STATUS
approved
A319171 Square array, read by antidiagonals, upwards: T(n,k) is the number of groups of order prime(k+1)^n. +10
0
1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 14, 5, 2, 1, 1, 51, 15, 5, 2, 1, 1, 267, 67, 15, 5, 2, 1, 1, 2328, 504, 77, 15, 5, 2, 1, 1, 56092, 9310, 684, 83, 15, 5, 2, 1, 1, 10494213, 1396077, 34297, 860, 87, 15, 5, 2, 1, 1, 49487367289, 5937876645 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
In 1960, Higman conjectured that the function f(n,p) giving the number of groups of prime-power order p^n, for fixed n and varying p, is a "Polynomial in Residue Classes" (PORC), i.e., there exist an integer M and polynomials q_i(x) in Z[x] (i = 1, 2, ..., M) such that if p = i mod M, then f(n,p) = q_i(p). The conjecture is confirmed for n <= 7.
LINKS
Table of n, a(n) for n=0..56.
H. U. Besche, B. Eick, and E. A. O'Brien. A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
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
Groupprops, Groups of prime power order
Groupprops, Higman's PORC conjecture
Groupprops, PORC function
Graham Higman, Enumerating p-Groups. I: Inequalities, Proc. London Math. Soc. Vol. 10 (1960), 24-30.
Graham Higman, Enumerating p-Groups. II: Problem whose solution is PORC, Proc. London Math. Soc. Vol. 10 (1960), 566-582.
Eamonn O'Brien, Polycyclic groups
Gordon Royle, Numbers of Small Groups
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 E. Witty, Enumeration of groups of prime-power order, PhD thesis, 2006.
Index entries for sequences related to groups
FORMULA
T(n,0) = A000679(n).
T(n,1) = A090091(n).
T(n,2) = A090130(n).
T(n,3) = A090140(n).
T(0,n) = 1, T(1,n) = 1, T(2,n) = 2 and T(3,n) = 5.
T(4,0) = 14 and T(4,n) = 15, n > 0.
T(5,n) = A232105(n+1).
T(6,n) = A232106(n+1).
T(7,n) = A232107(n+1).
EXAMPLE
Array begins:
(p = 2) (p = 3) (p = 5) (p = 7) (p = 11) (p = 13) ...
1 1 1 1 1 1 ...
1 1 1 1 1 1 ...
2 2 2 2 2 2 ...
5 5 5 5 5 5 ...
14 15 15 15 15 15 ...
51 67 77 83 87 97 ...
267 504 684 860 1192 1476 ...
2328 9310 34297 113147 750735 1600573 ...
...
MAPLE
with(GroupTheory): T:=proc(n, k) NumGroups(ithprime(k+1)^n); end proc: seq(seq(T(n-k, k), k=0..n), n=0..10); # Muniru A Asiru, Oct 03 2018
MATHEMATICA
(* This program uses Higman's PORC functions to compute the rows 0 to 7 *)
f[0, p_] := 1; f[1, p_] := 1; f[2, p_] := 2; f[3, p_] := 5;
f[4, p_] := If[p == 2, 14, 15];
f[5, p_] := If[p == 2, 51, If[p == 3, 67, 61 + 2*p + 2*GCD[p - 1, 3] + GCD[p - 1, 4]]];
f[6, p_] := If[p == 2, 267, If[p == 3, 504, 3*p^2 + 39*p + 344 + 24*GCD[p - 1, 3] + 11*GCD[p - 1, 4] + 2*GCD[p - 1, 5]]];
f[7, p_] := If[p == 2, 2328, If[p == 3, 9310, If[p == 5, 34297, 3*p^5 + 12*p^4 + 44*p^3 + 170*p^2 + 707*p + 2455 + (4*p^2 + 44*p + 291)*GCD[p - 1, 3] + (p^2 + 19*p + 135)*GCD[p - 1, 4] + (3*p + 31)*GCD[p - 1, 5] + 4*GCD[p - 1, 7] + 5*GCD[p - 1, 8] + GCD[p - 1, 9]]]];
tabl[kk_] := TableForm[Table[f[n, Prime[k+1]], {n, 0, 7}, {k, 0, kk}]];
PROG
(GAP) # This program computes the first 45 terms, rows 0..8.
P:=Filtered([1..300], IsPrime);;
T1:=List([0..7], n->List([0..15], k->NumberSmallGroups(P[k+1]^n)));;
T2:=[Flat(Concatenation(List([8], n->List([0], k->NumberSmallGroups(P[k+1]^n))), List([1..14], i->0)))];;
T:=Concatenation(T1, T2);;
b:=List([2..10], n->OrderedPartitions(n, 2));;
a:=Flat(List([1..Length(b)], i->List([1..Length(b[i])], j->T[b[i][j][2]][b[i][j][1]]))); # Muniru A Asiru, Oct 01 2018
CROSSREFS
Inspired by A158106.
Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232106, A232107.
KEYWORD
tabl,nonn,hard,more
AUTHOR
Franck Maminirina Ramaharo, Sep 12 2018
EXTENSIONS
a(55)=T(10,0) corrected by David Burrell, Jun 07 2022
a(56)=T(9,1) from David Burrell, Sep 01 2023
STATUS
approved
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Last modified August 30 03:24 EDT 2024. Contains 375523 sequences. (Running on oeis4.)