A232106 -id:A232106 - cp's OEIS frontend

A232105 Number of groups of order prime(n)^5.

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

Offset: 1

Eric M. Schmidt, Nov 21 2013

nonn

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

Cf. A000001, A000679, A090091, A090130, A090140, A232106, A232107.

Cf. A050997.

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

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)

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).

A232107 Number of groups of order prime(n)^7.

Original entry on oeis.org

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

Offset: 1

Eric M. Schmidt, Nov 21 2013

nonn

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

Cf. A000001, A000679, A090091, A090130, A090140, A232105, A232106.

Cf. A092759.

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

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

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)

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).

A269748 a(n) = 2p+61+2gcd(p-1,3)+gcd(p-1,4), where p = prime(n).

Original entry on oeis.org

68, 71, 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, 567, 581, 591, 605, 611, 625, 629

Offset: 1

N. J. A. Sloane, Mar 22 2016

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

[2*p+61 +2*Gcd(p-1,3)+Gcd(p-1,4): p in PrimesUpTo(700)]; // Vincenzo Librandi, Mar 26 2016

f1:=proc(n) local p; p:=ithprime(n);
2*p+61+2*gcd(p-1,3)+gcd(p-1,4);
end;

Table[2 Prime[n] + 61 + 2 GCD[Prime[n] - 1, 3] + GCD[Prime[n] -1, 4], {n, 60}] (* Vincenzo Librandi, Mar 26 2016 *)

a(n) = A232106(n) for n>=3. - R. J. Mathar, Jun 21 2025

A269749 a(n) = 3p^2+39p+344+24gcd(p-1,3)+11gcd(p-1,4)+2*gcd(p-1,5), where p = prime(n).

Original entry on oeis.org

471, 536, 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, 103496, 105812, 117292, 119736

Offset: 1

N. J. A. Sloane, Mar 22 2016

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

[3*p^2 + 39*p + 344 + 24*Gcd(p-1, 3) + 11*Gcd(p-1, 4) + 2*Gcd(p-1, 5): p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 26 2016

f2:=proc(n) local p; p:=ithprime(n);
3*p^2+39*p+344+24*gcd(p-1,3)+11*gcd(p-1,4)+2*gcd(p-1,5);
end;
[seq(f2(n),n=1..60)];

Table[3 Prime[n]^2 + 39 Prime[n] + 344 + 24 GCD[Prime[n] - 1, 3]+ 11 GCD[Prime[n] - 1, 4] + 2 GCD[Prime[n] - 1, 5], {n, 45}] (* Vincenzo Librandi, Mar 26 2016 *)

a(n) = A232106(n), n>2. - R. J. Mathar, May 23 2016

A271664 Erroneous version of A271811 (but for odd primes only).

Original entry on oeis.org

491, 668, 844, 1183, 1474, 1961, 2293, 2936, 4190, 4686, 6244, 7363, 7999, 9266, 11456, 13835, 14766, 17449, 19348, 20419, 23578, 25781, 29375, 34549, 37228, 38644, 41471, 43018, 46001, 57454, 60913, 66371, 68263, 77960, 80016, 86254, 92689, 97076, 103946, 111005, 113496

Offset: 2

Michel Marcus, Apr 12 2016

nonn

Previous name was "Number of non-abelian groups of order prime(n)^6".

Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.
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

Cf. A000001 (groups), A060689 (non-abelian groups),
Cf. A232106, A271664.
Cf. A030516 (primes^6)
Cf. A271811.

a(n) = if (n==2, 491, my(p=prime(n)); (13*p^2 + 145*p + 1338 + 80*gcd(p-1, 3) + 45*gcd(p-1, 4) + 8*gcd(p-1, 5) + 8*gcd(p-1, 6))/4);

a(n) = (13*p^2 + 145*p + 1338 + 80*gcd(p-1,3) + 45*gcd(p-1,4) + 8*gcd(p-1, 5) + 8*gcd(p-1,6))/4 for n>2 and where p = prime(n). See [Rodney James].

A271811 Number of non-abelian groups of order prime(n)^6.

Original entry on oeis.org

256, 493, 673, 849, 1181, 1465, 1933, 2253, 2865, 4057, 4529, 6001, 7053, 7653, 8841, 10897, 13125, 14001, 16509, 18281, 19285, 22233, 24285, 27637, 32461, 34953, 36273, 38901, 40345, 43117, 53769, 56981, 62053, 63813, 72817, 74729, 80521, 86493, 90561, 96937, 103485, 105801, 117281

Offset: 1

Altug Alkan, Apr 14 2016

nonn

A000688(p^6) is 11 for all prime p.

Muniru A Asiru, Table of n, a(n) for n = 1..1229
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

Cf. A000688, A030516, A060689, A232106, A271664.

A271811 := Concatenation([256, 493], List(Filtered([5..10^4], IsPrime), p -> 3 * p^2 + 39 * p + 333 + 24 * Gcd(p-1, 3) + 11 * Gcd(p-1, 4) + 2 * Gcd(p-1,5))); # Muniru A Asiru, Nov 18 2017

Table[FiniteGroupCount[#] - FiniteAbelianGroupCount[#] &[Prime[n]^6], {n, 43}] (* Michael De Vlieger, Apr 15 2016, after Vladimir Joseph Stephan Orlovsky at A060689 *)

a(n) = if (n==1, 256, if (n==2, 493, my(p=prime(n)); 3*p^2 + 39*p + 333 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5)));

a(n) = A232106(n) - 11.
a(n) = A060689(prime(n)^6) = A060689(A030516(n)).
For a prime p > 3, the number of non-abelian groups of order p^6 is 3p^2 + 39p + 333 + 24 gcd(p - 1, 3) + 11 gcd(p - 1, 4) + 2 gcd(p - 1, 5).

A319171 Square array, read by antidiagonals, upwards: T(n,k) is the number of groups of order prime(k+1)^n.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Sep 12 2018

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.

			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  ...
     ...

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

Inspired by A158106.

Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232106, A232107.

# 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

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

(* 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}]];

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).

a(55)=T(10,0) corrected by David Burrell, Jun 07 2022

a(56)=T(9,1) from David Burrell, Sep 01 2023

cp's OEIS Frontend

A232105 Number of groups of order prime(n)^5.

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A232107 Number of groups of order prime(n)^7.

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A269748 a(n) = 2*p+61+2*gcd(p-1,3)+gcd(p-1,4), where p = prime(n).

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A269749 a(n) = 3*p^2+39*p+344+24*gcd(p-1,3)+11*gcd(p-1,4)+2*gcd(p-1,5), where p = prime(n).

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A271664 Erroneous version of A271811 (but for odd primes only).

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A271811 Number of non-abelian groups of order prime(n)^6.

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A319171 Square array, read by antidiagonals, upwards: T(n,k) is the number of groups of order prime(k+1)^n.

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A269748 a(n) = 2p+61+2gcd(p-1,3)+gcd(p-1,4), where p = prime(n).

A269749 a(n) = 3p^2+39p+344+24gcd(p-1,3)+11gcd(p-1,4)+2*gcd(p-1,5), where p = prime(n).