A054397 -id:A054397 - cp's OEIS frontend

A298910 Numbers m such that there are precisely 19 groups of order m.

1029, 5145, 6591, 7803, 8001, 11319, 11739, 12789, 17157, 17493, 20577, 21567, 23667, 23877, 27993, 31311, 32955, 33411, 34671, 34713, 39015, 39753, 40005, 42189, 42861, 45675, 47691, 48363, 49833

Offset: 1

Muniru A Asiru, Jan 28 2018

nonn

			For m = 1029, the 19 groups are C1029, C147 x C7, C3 x ((C7 x C7) : C7), C3 x (C49 : C7), C21 x C7 x C7, C343 : C3, C49 x (C7 : C3), C7 x (C49 : C3), (C49 x C7) : C3, (C49 x C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, (C49 : C7) : C3, C7 x ((C7 x C7) : C3), C7 x ((C7 x C7) : C3), (C7 x C7 x C7) : C3, (C7 x C7 x C7) : C3, C7 x C7 x (C7 : C3) where C means the Cyclic group of the stated order and the symbols x and : mean direct and semidirect products respectively.

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.
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), 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), this sequence (k=19), A298911 (k=20).

with(GroupTheory):
for n from 1 to 3*10^5 do if NumGroups(n) = 19 then print(n); fi; od;

Sequence is { m | A000001(m) = 19 }.

Shortened to remove possibly incorrect terms by Andrew Howroyd, Jan 28 2022

A143928 2*p^2, for p an odd prime.

Original entry on oeis.org

18, 50, 98, 242, 338, 578, 722, 1058, 1682, 1922, 2738, 3362, 3698, 4418, 5618, 6962, 7442, 8978, 10082, 10658, 12482, 13778, 15842, 18818, 20402, 21218, 22898, 23762, 25538, 32258, 34322, 37538, 38642, 44402, 45602, 49298, 53138, 55778, 59858

Offset: 1

Jonathan Vos Post, Sep 05 2008

For these numbers m, there are precisely 5 groups of order m, hence it is a subsequence of A054397. The 5 groups are C_{2*p^2}, C_2 X (C_p X C_p), C_p^2 : C_2 ~ D_{2*p^2}, and two non-isomorphic groups (C_p X C_p) : C_2, where C, D mean cyclic, dihedral groups of the stated order; the symbols ~, X and : mean isomorphic to, direct and semidirect products respectively. - Bernard Schott, Dec 10 2021

			a(1) = 2*A065091(1)^2 = 2*3^2 = 18.
a(2) = 2*A065091(2)^2 = 2*5^2 = 50.
a(3) = 2*A065091(3)^2 = 2*7^2 = 98.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Michael Hilgemann and Siu-Hung Ng, Hopf algebras of dimension 2p^2, arXiv:0809.0699 [math.QA], 2008.

Subsequence of A079704.

Cf. A054397, A065091.

2#^2&/@Prime[Range[2,40]] (* Harvey P. Dale, Jul 23 2021 *)

from sympy import prime
def a(n): return 2*prime(n+1)**2
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Dec 10 2021

a(n) = A079704(n+1) for n>0.

A295991 Numbers n such that there are precisely 5 groups of orders n and n + 1.

Original entry on oeis.org

1107, 1268, 2036, 2163, 2403, 2451, 2612, 3075, 3284, 3411, 3698, 4052, 4388, 4772, 4868, 5156, 5715, 6212, 6452, 6771, 6788, 7011, 7155, 8547, 8612, 8643, 8948, 9092, 9124, 10227, 11204, 11444, 11636, 11811, 13652, 13778, 14067, 14324, 14547, 17043, 17427, 18818, 18915, 19892

Offset: 1

Muniru A Asiru, Dec 02 2017

nonn

Equivalently, lower member of consecutive terms of A054397.

			1107 is in the sequence because A000001(1107) = A000001(1108) = 5, 1268 is in the sequence because A000001(1268) = A000001(1269) = 5 and 2163 is in the sequence because A000001(2163) = A000001(2164) = 5.

Muniru A Asiru, Table of n, a(n) for n = 1..103
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.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001, A054397.

Sequence is { n | A000001(n) = 5, A000001(n+1) = 5 }.

A349495 Numbers p^2*q, p (2,3).

Original entry on oeis.org

28, 44, 63, 76, 92, 117, 124, 172, 188, 236, 268, 275, 279, 284, 316, 332, 387, 412, 428, 508, 524, 549, 556, 603, 604, 652, 668, 711, 716, 764, 775, 796, 844, 873, 892, 908, 927, 956, 1004, 1025, 1052, 1084, 1132, 1228, 1244, 1251, 1324, 1359, 1388, 1413, 1421

Offset: 1

Bernard Schott, Dec 15 2021

nonn

For these terms m, there are precisely 4 groups of order m, so this is a subsequence of A054396.
Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p, and the two others that are nonabelian are C_q : (C_p x C_p), and C_q : C_p^2. Note that when p = 2, C_q : (C_p x C_p) ~ D_{p^2*q}. Here C and D mean cyclic and dihedral groups of the stated order, the symbols ~,  X and : mean "isomorphic to", direct and semidirect products respectively.
Why (p,q) <> (2,3)?  Because there are 5 groups of order 12, and in this particular case, the 5th group is the alternating group A_4 because 2^2*3 = 4!/2 (see Example section in A054397).
Contains 4*r for r in A002145 and r > 3. - Alois P. Heinz, Dec 15 2021

			28 = 2^2*7, 2 divides 7-1 = 6 and 2^2 does not divide 7-1 = 6, hence 28 is a term.
63 = 3^2*7, 3 divides 7-1 = 6 and 3^2 does not divide 7-1 = 6, hence 63 is another term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Intersection of A054396 and A054753.

Cf. A002145.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && IntegerExponent[p[[2]] - 1, p[[1]]] == 1]; Select[Range[28, 1500], q]  (* Amiram Eldar, Dec 16 2021 *)

isok(m) = if (m==12, return(0)); my(f=factor(m)); if (f[,2] == [2,1]~, my(p=f[1,1], q=f[2,1]); (((q-1) % p) == 0) && (((q-1) % p^2) != 0);); \\ Michel Marcus, Dec 16 2021

from sympy import factorint
def ok(n):
    if n < 13: return False
    f = factorint(n)
    sig, p, q = list(f.values()), min(f), max(f)
    return sig == [2, 1] and (q-1)%p == 0 and (q-1)%p**2 != 0
print([m for m in range(1422) if ok(m)]) # Michael S. Branicky, Dec 16 2021

More terms from Alois P. Heinz, Dec 15 2021

A350115 Numbers p^2*q, p

Original entry on oeis.org

20, 52, 68, 116, 148, 164, 171, 212, 244, 292, 333, 356, 388, 404, 436, 452, 548, 596, 628, 657, 692, 724, 772, 788, 916, 932, 964, 981, 1028, 1076, 1108, 1124, 1143, 1172, 1252, 1268, 1348, 1396, 1412, 1467, 1492, 1556, 1588, 1604, 1629, 1636, 1684, 1732, 1791, 1796, 1828, 1844

Offset: 1

Bernard Schott, Dec 14 2021

nonn

For these terms m, there are precisely 5 groups of order m, so this is a subsequence of A054397.

Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p = C_q X (C_p)^2, and the three others that are nonabelian are C_q : (C_p x C_p), and two nonisomorphic semi-direct products C_q : C_p^2. Here C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.

			20 = 2^2*5 and 2^2 divides 5-1, hence 20 is a term.
171 = 3^2*19 and 3^2 divides 19-1, hence 171 is another term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Other subsequences of A054397: A030078, A079704, A143928.

Subsequence of A054753.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && Divisible[p[[2]] - 1, p[[1]]^2]]; Select[Range[2000], q] (* Amiram Eldar, Dec 14 2021 *)

isok(m) = {my(f=factor(m)); if (f[,2] == [2,1]~, my(p=f[1,1], q=f[2,1]); ((q-1) % p^2) == 0;);} \\ Michel Marcus, Dec 14 2021

from sympy import integer_nthroot, isprime, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit, 4)[0]
    for p in primerange(1, maxp+1):
        m = p**2
        for t in range(m, limit//m, m):
            if isprime(t+1):
                aset.add(p**2*(t+1))
    return sorted(aset)
print(aupto(1844)) # Michael S. Branicky, Dec 14 2021

More terms from Michel Marcus, Dec 14 2021

A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 14, 33

Offset: 0

Des MacHale, Feb 26 2021

The number of groups of order 2^n is A000679(n); the percentage of the 2-groups which occur as automorphism groups appears to decrease as n increases: 100, 100, 100, 60, 28.5, 17.6, 5.2, 1.4, ...

Jianing Song remarks that it is also interesting to consider infinite groups, and asks if there is an infinite group G with Aut(G) isomorphic to C_8. Des MacHale, Mar 03 2021, replies that at present this is not known. [Comment edited by N. J. A. Sloane, Mar 07 2021]

			a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
From _Bernard Schott_, Feb 26 2021: (Start)
Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
There exist five groups of order 8 (A054397), the three groups C_4 x C_2, D_4, C_2 x C_2 x C_2 occur as automorphim groups of order 8, but the cyclic group C_8 and the quaternions group Q_8 never occur as Aut(G) for some finite G, so a(3) = 3. (End)

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A000679, A054397, A340521, A341823, A341825.

a(n) <= A000679(n). - Des MacHale, Mar 02 2021

Offset modified by Jianing Song, Mar 02 2021

A384370 Squarefree integers m such that there are precisely 5 groups of order m.

Original entry on oeis.org

273, 399, 651, 741, 777, 1209, 1281, 1365, 1407, 1443, 1533, 1659, 1677, 1767, 1995, 2037, 2109, 2163, 2289, 2379, 2451, 2613, 2847, 2919, 3003, 3171, 3297, 3423, 3441, 3477, 3705, 3783, 3801, 3819, 3885, 3999, 4017, 4053, 4161, 4179, 4251, 4389, 4503, 4641, 4683, 4773, 4809, 4953

Offset: 1

Robin Jones, May 27 2025

nonn

These are precisely the squarefree integers m such that 3|m, there are exactly two prime factors of m which are congruent to 1 modulo 3, and there are no other relations of the form p = 1 mod q for any pair of prime factors p, q of m.
This is a subsequence of A054397.
This sequence is infinite.

			273 is in this sequence as 273 is squarefree, and A000001(273) = 5.

Robin Jones, Table of n, a(n) for n = 1..10000

Cf. A054397.

Select[Range[5000], SquareFreeQ[#]&&FiniteGroupCount[#] == 5 &] (* James C. McMahon, May 31 2025 *)

cp's OEIS Frontend

A298910 Numbers m such that there are precisely 19 groups of order m.

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A143928 2*p^2, for p an odd prime.

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A295991 Numbers n such that there are precisely 5 groups of orders n and n + 1.

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A349495 Numbers p^2*q, p (2,3).

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A350115 Numbers p^2*q, p

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A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

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A384370 Squarefree integers m such that there are precisely 5 groups of order m.

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Mathematica