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1
1, 2, 5, 15, 67, 504, 9310
COMMENTS
Previous name was: Marcus Du Sautoy's sequences of symmetries from Higman's "Polynomial on residue classes".
REFERENCES
Marcus Du Sautoy, Symmetry: A Journey into the Patterns of Nature,Harper (March 11, 2008),page 96
Number of groups of order 5^n.
+10
8
1, 1, 2, 5, 15, 77, 684, 34297
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.
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), ...
AUTHOR
Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004
EXTENSIONS
Corrected and extended by Eamonn O'Brien, Mar 06 2010
Number of groups of order 7^n.
+10
7
1, 1, 2, 5, 15, 83, 860, 113147
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.
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), ...
AUTHOR
Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004
EXTENSIONS
Updated reference for p^7 Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Mar 06 2010
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
COMMENTS
Isomorphism types of groups and nilpotent Lie rings with order prime(n)^6.
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)):
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
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
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
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
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)):
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
a(n) is the smallest composite number coprime to n.
+10
3
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
COMMENTS
If n is the n-th primorial, then a(n) = prime(n+1)^2.
FORMULA
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
EXAMPLE
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.
MATHEMATICA
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}]
Number of exponent-3 class 2 groups of order 3^n.
+10
3
0, 0, 1, 3, 7, 24, 103, 1565, 602419, 4896600938, 5876589263966179
LINKS
Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
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
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.
FORMULA
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.
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
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