A090091 -id:A090091 - cp's OEIS frontend

A158079 Duplicate of A090091.

Original entry on oeis.org

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

Offset: 1

dead

Previous name was: Marcus Du Sautoy's sequences of symmetries from Higman's "Polynomial on residue classes".

Marcus Du Sautoy, Symmetry: A Journey into the Patterns of Nature,Harper (March 11, 2008),page 96

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

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

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

A232106 Number of groups of order prime(n)^6.

Original entry on oeis.org

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

Offset: 1

Eric M. Schmidt, Nov 21 2013

nonn

Isomorphism types of groups and nilpotent Lie rings with order prime(n)^6.

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

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

Cf. A030516.

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

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

Table[FiniteGroupCount[Prime[n]^6], {n, 40}] (* Michael De Vlieger, Apr 12 2016 *)

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

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)

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).

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

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

Original entry on oeis.org

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).

A089090 a(n) is the smallest composite number coprime to n.

Original entry on oeis.org

4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 9, 4, 25, 4

Offset: 1

Labos Elemer, Nov 26 2003

If n is the n-th primorial, then a(n) = prime(n+1)^2.

			n=30: below 30 coprimes to 30 phi(30)=8 numbers are relevant but each 1 or primes; so a(8)>30; the first suitable number is a(30)=49.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A053669, A007978, A053670-A053674, A090091-A090093.

m=0;Table[fla=1;Do[s=GCD[n, k];If[Equal[s, 1]&&!PrimeQ[n]&&!Equal[n, 1]&& Equal[fla, 1], m=m+1;Print[n];fla=0], {n, 1, 130}], {k, 1, 256}]

A089090(n) = forprime(p=2, , if(n%p, return(p*p))); \\ Antti Karttunen, Dec 19 2018

a(n) = A053669(n)^2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} ((p^2*(p-1)/Product_{q prime <= p} q)) = 10.3344588090... . - Amiram Eldar, Jul 25 2022

Offset corrected by Antti Karttunen, Dec 19 2018

A252785 Number of exponent-3 class 2 groups of order 3^n.

Original entry on oeis.org

0, 0, 1, 3, 7, 24, 103, 1565, 602419, 4896600938, 5876589263966179

Offset: 0

Eric M. Schmidt, Dec 21 2014

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.

Cf. A090091, A252784, A252786, A252787.

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

A158079 Duplicate of A090091.

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A090130 Number of groups of order 5^n.

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A232106 Number of groups of order prime(n)^6.

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A090140 Number of groups of order 7^n.

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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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A089090 a(n) is the smallest composite number coprime to n.

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A252785 Number of exponent-3 class 2 groups of order 3^n.

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