A298910 -id:A298910 - cp's OEIS frontend

A054395 Numbers m such that there are precisely 2 groups of order m.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 45, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94, 99, 105, 106, 111, 118, 121, 122, 129, 134, 142, 146, 153, 155, 158, 165, 166, 169, 175, 178, 183, 194, 195, 201, 202, 203, 205, 206, 207, 214, 218, 219, 226, 231, 237

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

Givens characterizes this sequence, see Theorem 5. In particular, this sequence is ({n: A215935(n) = 1} INTERSECT A005117) UNION (A060687 INTERSECT A051532). - Charles R Greathouse IV, Aug 27 2012 [This is now A350586 UNION A350322. - Charles R Greathouse IV, Jan 08 2022]

Numbers m such that A000001(m) = 2. - Muniru A Asiru, Nov 03 2017

			For m = 4, the 2 groups of order 4 are C4, C2 x C2; for m = 6, the 2 groups of order 6 are S3, C6; and for m = 9, the 2 groups of order 9 are C9, C3 x C3 where C is the cyclic group of the stated order and S is the symmetric group of the stated degree. The symbol x means direct product. - _Muniru A Asiru_, Oct 24 2017

Muniru A Asiru and Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..422 from Muniru A Asiru.
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Clint Givens, Orders for which there exist exactly two groups (2006)
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Equals A350586 UNION A350322.

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: this sequence (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A054395 := Filtered([1..2015], n -> NumberSmallGroups(n) = 2); # Muniru A Asiru, Oct 24 2017

IsGivensInt := function(n)
  local p, f; p := GcdInt(n, Phi(n));
  if not IsPrimeInt(p) then return false; fi;
  if n mod p^2 = 0 then return 1 = GcdInt(p+1, n); fi;
  f := PrimePowersInt(n);
  return 1 = Number([1..QuoInt(Length(f),2)], k->f[2*k-1] mod p = 1);
end;;
Filtered([1..240], IsGivensInt); # Gheorghe Coserea, Dec 04 2017

Select[Range[240], FiniteGroupCount[#] == 2&]
(* or: *)
okQ[n_] := Module[{p, f}, p = GCD[n, EulerPhi[n]]; If[! PrimeQ[p], Return[False]]; If[Mod[n, p^2] == 0, Return[1 == GCD[p + 1, n]]]; f = FactorInteger[n]; 1 == Sum[Boole[Mod[f[[k, 1]], p] == 1], {k, 1, Length[f]}]];
Select[Range[240], okQ] (* Jean-François Alcover, Dec 08 2017, after Gheorghe Coserea *)

is(n) = {
  my(p=gcd(n,eulerphi(n)), f);
  if (!isprime(p), return(0));
  if (n%p^2 == 0, return(1 == gcd(p+1, n)));
  f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k,1]%p==1);
};
seq(N) = {
  my(a = vector(N), k=0, n=1);
  while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(58) \\ Gheorghe Coserea, Dec 03 2017

More terms from Christian G. Bower, May 25 2000

A054397 Numbers m such that there are precisely 5 groups of order m.

Original entry on oeis.org

8, 12, 18, 20, 27, 50, 52, 68, 98, 116, 125, 135, 148, 164, 171, 212, 242, 244, 273, 292, 297, 333, 338, 343, 356, 388, 399, 404, 436, 452, 459, 548, 578, 596, 621, 628, 651, 657, 692, 722, 724, 741, 772, 777, 783, 788, 825, 855, 875, 916, 932, 964, 981

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

For m = 2*p^2 (p prime), there are precisely 5 groups of order m, so A079704 and A143928 (p odd prime) are two subsequences. - Bernard Schott, Dec 10 2021
For m = p^3, p prime, there are also 5 groups of order m, so A030078, where these groups are described, is another subsequence. - Bernard Schott, Dec 11 2021
For m squarefree, there are 5 groups of order m if and only if all of the following hold: 3|m, there are exactly two prime factors p,q of m such that p,q = 1 mod 3, no other relations of the form p' = 1 mod q' hold for p',q' prime factors of m. - Robin Jones, May 27 2025

			For m = 8, the 5 groups of order 8 are C8, C4 x C2, D8, Q8, C2 x C2 x C2 and for m = 12 the 5 groups of order 12 are C3 : C4, C12, A4, D12, C6 x C2 where C, D, Q  mean cyclic, dihedral, quaternion groups of the stated order and A is the alternating group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 03 2017

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..2035, (terms 1..120 from Muniru A Asiru and Georg Fischer).
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), this sequence (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

Cf. A384370 (squarefree numbers in this sequence).

A054397 := Filtered([1..2015], n -> NumberSmallGroups(n) = 5); # Muniru A Asiru, Nov 03 2017

Select[Range[10^4], FiniteGroupCount[#] == 5 &] (* Robert Price, May 23 2019 *)

Sequence is { k | A000001(k) = 5 }. - Muniru A Asiru, Nov 03 2017

More terms from Christian G. Bower, May 25 2000

A055561 Numbers m such that there are precisely 3 groups of order m.

Original entry on oeis.org

75, 363, 609, 867, 1183, 1265, 1275, 1491, 1587, 1725, 1805, 2067, 2175, 2373, 2523, 3045, 3525, 3685, 3795, 3975, 4137, 4205, 4335, 4425, 4895, 5019, 5043, 5109, 5901, 5915, 6171, 6225, 6627, 6675, 6699, 7935, 8025, 8427, 8475, 8855, 9429, 9537, 10275

Offset: 1

Christian G. Bower, May 25 2000; Nov 12 2003; Feb 17 2006

nonn

Let gnu(n) (= A000001(n)) denote the "group number of n" defined in A000001 or in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), then the sequence n -> gnu(a(n)) -> gnu(gnu(a(n))) consists of 1's. - Muniru A Asiru, Nov 19 2017
From Jianing Song, Dec 05 2021: (Start)
Contains all numbers of the form k = p*q^2, where p, q are odd primes such that q == -1 (mod p) (see A350245). The 3 groups are C_(p*q^2), C_q X C_(p*q) and (C_q X C_q) : C_p, where : means semidirect product. The third group, which is the only non-abelian group of order k, can be constructed as follows: in F_q the polynomial x^(p-1) + x^(p-2) + ... + x + 1 factors into quadratic polynomials. Pick one factor x^2 + a*x + b (all factors give the same group), then (C_q X C_q) : C_p has representation .
It seems that all terms are odd. (End)

			For m = 75, the 3 groups of order 75 are C75, (C5 x C5) : C3, C15 x C5 and for m = 363 the 3 groups of order 363 are C363, (C11 x C11) : C3, C33 x C11 where C is the Cyclic group of the stated order. The symbols x and : mean direct and semi-direct products respectively. - _Muniru A Asiru_, Oct 24 2017

Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..206 from Muniru A Asiru.
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
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.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Gordon Royle, Numbers of Small Groups [dead link]
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), this sequence (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A350245 is a subsequence.

is(n) = {
  my(p = gcd(n, eulerphi(n)),f,g);
  if (isprime(p), return(n % p^2 == 0 && isprime(gcd(p+1, n))));
  if (omega(p) != 2 || !issquarefree(n), return(0));
  f = factor(n); g = factor(p);
  1 == g[2,1] % g[1,1] &&
  1 == sum(k=1, matsize(f)[1], f[k,1] % g[1,1] == 1) &&
  1 == sum(k=1, matsize(f)[1], f[k,1] % g[2,1] == 1);
};
seq(N) = {
  my(a = vector(N), k=0, n=1);
  while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(43) \\ Gheorghe Coserea, Dec 12 2017

A054396 Numbers m such that there are precisely 4 groups of order m.

Original entry on oeis.org

28, 30, 44, 63, 66, 70, 76, 92, 102, 117, 124, 130, 138, 154, 170, 172, 174, 182, 188, 190, 230, 236, 238, 246, 266, 268, 275, 279, 282, 284, 286, 290, 315, 316, 318, 322, 332, 354, 370, 374, 387, 412, 418, 426, 428, 430, 434, 442, 465, 470, 494, 495, 498

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

			For m = 28, the 4 groups of order 8 are C7 : C4, C28, D28, C14 x C2 and for m = 30 the 4 groups of order 30 are C5 x S3, C3 x D10, D30, C30 where C, D mean cyclic, dihedral groups of the stated order and S is the symmetric group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 04 2017

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..10000 (terms 1..369 from Muniru A Asiru).
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), this sequence (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A054396 := Filtered([1..2015], n -> NumberSmallGroups(n) = 4); # Muniru A Asiru, Nov 04 2017

Select[Range[500], FiniteGroupCount[#] == 4 &] (* Jean-François Alcover, Dec 08 2017 *)

Sequence is { m | A000001(m) = 4 }. - Muniru A Asiru, Nov 04 2017

More terms from Christian G. Bower, May 25 2000

A135850 Numbers m such that there are precisely 6 groups of order m.

Original entry on oeis.org

42, 78, 110, 114, 147, 186, 222, 225, 258, 310, 366, 402, 406, 410, 438, 474, 506, 507, 525, 582, 602, 610, 618, 654, 710, 735, 762, 834, 906, 942, 975, 978, 994, 1010, 1083, 1086, 1089, 1158, 1194, 1266, 1310, 1338, 1374, 1378, 1425, 1446, 1474, 1510, 1582

Offset: 1

N. J. A. Sloane, based on a suggestion from Neven Juric, Mar 08 2008

nonn

Let gnu(n) = A000001(n) denote the "group number of n" defined in A000001 or in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), then the sequence n -> gnu(a(n)) -> gnu(gnu(a(n))) -> gnu(gnu(gnu(a(n)))) consists of 1's. - Muniru A Asiru, Nov 19 2017

			For m = 42, the 6 groups of order 42 are (C7 : C3) : C2, C2 x (C7 : C3), C7 x S3, C3 x D14, D42, C42 and for n = 78 the 6 groups of order 78 are (C13 : C3) : C2, C2 x (C13 : C3), C13 x S3, C3 x D26, D78, C78 where C, D mean Cyclic, Dihedral groups of the stated order and S is the Symmetric group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 04 2017

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..1099 (terms 1..91 from Muniru A Asiru).
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), this sequence (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A135850 := Filtered([1..2015], n -> NumberSmallGroups(n) = 6); # Muniru A Asiru, Nov 04 2017

Select[Range[10^4], FiniteGroupCount[#] == 6 &] (* Robert Price, May 23 2019 *)

Sequence is { m | A000001(m) = 6 }. - Muniru A Asiru, Nov 04 2017

A249551 Numbers m such that there are precisely 8 groups of order m.

Original entry on oeis.org

510, 690, 870, 910, 1122, 1190, 1330, 1395, 1410, 1590, 1610, 1770, 1914, 2002, 2210, 2346, 2470, 2490, 2590, 2618, 2670, 2706, 2745, 2926, 2958, 2990, 3094, 3102, 3210, 3230, 3290, 3390, 3458, 3465, 3498, 3710, 3770, 3894, 3910, 4002, 4110, 4130, 4182, 4186, 4370, 4470

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..1064
H. U. Besche, B. Eick and E. A. O'Brien, Numbers of isomorphism types of finite groups of given order
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), this sequence (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249551 := Filtered([1..2015], n -> NumberSmallGroups(n) = 8); # Muniru A Asiru, Oct 18 2017

Select[Range[10^4], FiniteGroupCount[#] == 8 &] (* A current limit in Mathematica is such that some orders >2047 may not be evaluated.*) (* Robert Price, May 24 2019 *)

a(15)-a(16) from Muniru A Asiru, Oct 18 2017
More terms from Michael De Vlieger, Oct 18 2017
Missing terms added by Andrey Zabolotskiy, Oct 27 2017

A249552 Numbers m such that there are precisely 9 groups of order m.

Original entry on oeis.org

308, 532, 644, 836, 868, 1316, 1364, 1652, 1748, 1815, 1876, 1892, 2068, 2212, 2324, 2356, 2596, 2852, 2884, 2996, 3124, 3268, 3476, 3572, 3652, 3668, 3892, 3956, 4228, 4263, 4484, 4532, 4564, 4676, 4708, 5012, 5092, 5332, 5348, 5396, 5428, 5572, 5588, 5764, 5828, 6004, 6116, 6164, 6244, 6308, 6356, 6532

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Muniru A Asiru and Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..456
Max Horn, Numbers of isomorphism types of finite groups of given order
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), this sequence (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249552:=Filtered([1..2015],n->NumberSmallGroups(n)=9); # Muniru A Asiru, Nov 17 2017

select(t -> GroupTheory:-NumAbelianGroups(t) <= 9 and GroupTheory:-NumGroups(t) = 9, [$1..10000]); # Robert Israel, Mar 26 2018

a(13)-a(16) from Muniru A Asiru, Oct 21 2017
More terms from Muniru A Asiru, Oct 23 2017
More terms from Muniru A Asiru, Nov 17 2017
Incorrect b-file shortened by Jorge R. F. F. Lopes, Jan 07 2022

A249553 Numbers m such that there are precisely 10 groups of order m.

Original entry on oeis.org

90, 132, 198, 276, 306, 350, 414, 490, 522, 564, 650, 708, 738, 846, 850, 852, 950, 954, 996, 1062, 1078, 1150, 1274, 1278, 1284, 1450, 1485, 1494, 1572, 1602, 1666, 1690, 1694, 1818, 1850, 1862, 1926, 2004, 2034, 2148, 2150, 2254, 2292, 2325, 2350, 2358, 2466, 2475, 2650, 2682, 2724, 2868, 2890, 2950, 3006, 3012, 3038, 3114, 3146, 3156

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..799
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), this sequence (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249553 := Filtered([1..2015], n -> NumberSmallGroups(n) = 10); # Muniru A Asiru, Oct 16 2017

Select[ Range@2047, FiniteGroupCount@# == 10 &] (* Robert G. Wilson v, Nov 30 2017 *)

More terms from Michael De Vlieger, Oct 16 2017

More terms from Muniru A Asiru, Oct 24 2017

A249554 Numbers m such that there are precisely 11 groups of order m.

Original entry on oeis.org

140, 364, 380, 460, 476, 572, 748, 819, 860, 940, 988, 1036, 1148, 1180, 1196, 1276, 1292, 1340, 1484, 1564, 1580, 1612, 1628, 1660, 1708, 1804, 1953, 2044, 2060, 2108, 2140, 2204, 2236, 2332, 2444, 2492, 2540, 2668, 2684, 2716, 2780, 2812, 2828, 2924, 3052, 3068, 3116, 3196, 3212

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..929
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), this sequence (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249554 := Filtered([1..2015], n -> NumberSmallGroups(n) = 11); # Muniru A Asiru, Oct 16 2017

with(GroupTheory): select(n->NumGroups(n)=11,[$1..4000]); # Muniru A Asiru, Mar 28 2018

Select[Range[10^4], FiniteGroupCount[#] == 11 &] (* A current limit in Mathematica is such that some orders >2047 may not be evaluated.*)(* Robert Price, May 24 2019 *)

More terms added by Muniru A Asiru, Oct 23 2017

Incorrect b-file shortened by Andrew Howroyd, Jan 28 2022

A249555 Numbers m such that there are precisely 12 groups of order m.

Original entry on oeis.org

88, 152, 184, 196, 204, 210, 248, 330, 344, 348, 376, 390, 462, 472, 484, 492, 536, 568, 570, 632, 636, 664, 714, 770, 824, 856, 858, 966, 1016, 1048, 1068, 1110, 1112, 1208, 1212, 1230, 1254, 1290, 1304, 1326, 1336, 1356, 1430, 1432, 1444, 1518, 1528, 1592, 1644

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..1512
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), this sequence (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249555 := Filtered([1..2015], n -> NumberSmallGroups(n) = 12); # Muniru A Asiru, Oct 16 2017

Select[Range@ 2074, FiniteGroupCount@ # == 12 &] (* Michael De Vlieger, Oct 16 2017. Note: extending the range to 2075 and further will result in incorrect output. - Andrey Zabolotskiy, Oct 27 2017 *)

cp's OEIS Frontend

A054395 Numbers m such that there are precisely 2 groups of order m.

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A054397 Numbers m such that there are precisely 5 groups of order m.

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A055561 Numbers m such that there are precisely 3 groups of order m.

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A054396 Numbers m such that there are precisely 4 groups of order m.

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A135850 Numbers m such that there are precisely 6 groups of order m.

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A249551 Numbers m such that there are precisely 8 groups of order m.

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A249552 Numbers m such that there are precisely 9 groups of order m.

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