Robin Jones - cp's OEIS frontend

A384607 Possible values for the number of groups of order equal to a prime power, in order of size.

1, 2, 5, 14, 15, 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, 267, 269, 275, 279, 289, 293, 323, 327, 341, 347, 365, 371, 385, 395, 399, 413, 423, 433, 447, 457, 461, 467, 491, 504, 515

Offset: 1

Robin Jones, Jun 04 2025

nonn

Possible values of A098885, ordered by size.

This sequence is the same regardless of whether 1 is considered a prime power or not (see A000961 for discussion on this) as A000001(1)=A000001(p)=1 for all p.

			1 is in this sequence because A000001(2) = 1.
2 is in this sequence because A000001(2^2) = 2.
5 is in this sequence because A000001(2^3) = 5.
3 is not in this sequence as no prime power p^k has A000001(p^k)=3.

Robin Jones, Table of n, a(n) for n = 1..180

Cf. A000001, A098885, A384606.

A384606 Possible values for the number of groups of order equal to a prime power, in order of first appearance.

Original entry on oeis.org

1, 2, 5, 14, 51, 267, 15, 2328, 67, 56092, 10494213, 504, 49487367289

Offset: 1

Robin Jones, Jun 04 2025

Equal A098885 with the duplicate entries removed.
a(14) = A000001(2048) (this value is currently unknown).
This sequence is the same regardless of whether 1 is considered a prime power or not (see A000961 for discussion on this) as A000001(1) = A000001(p) = 1 for all p.

			1 is in this sequence because A000001(2) = 1.
2 is in this sequence because A000001(2^2) = 2.
5 is in this sequence because A000001(2^3) = 5.
3 is not in this sequence as no prime power p^k has A000001(p^k)=3.

Cf. A098885, A384607.

A384609 Possible values for the number of nilpotent groups of a finite order, ordered by size.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 15, 16, 20, 25, 28, 30, 32, 40, 50, 51, 56, 60, 64, 67, 70, 75, 77, 80, 83, 87, 97, 100, 101, 102, 107, 111, 112, 120, 125, 128, 131, 134, 140, 145, 149, 150, 154, 155, 159, 160, 166, 173, 174, 183, 193, 194, 196, 200, 202, 203, 204, 207

Offset: 1

Robin Jones, Jun 04 2025

nonn

A066060 sorted and duplicates removed.

List of all possible products of terms in A384607 (possibly with use of the same integer more than once).

			1 is in this sequence as there is exactly 1 nilpotent group of order 1.
2 is in this sequence as there are exactly 2 nilpotent groups of order 4.
4 is in this sequence as there are exactly 4 nilpotent groups of order 36.
3 is not in this sequence as there are never exactly 3 nilpotent groups of any given order.

Robin Jones, Table of n, a(n) for n = 1..135

Cf. A066060, A384607.

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 *)

A384146 Smallest squarefree order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.

Original entry on oeis.org

1, 6, 609, 30, 273, 42, 903, 510, 8729, 3255, 494711, 210, 16951, 5115, 54431, 1218

Offset: 1

Robin Jones, May 21 2025

It has been established that every n < 10000000 arises as the number of groups up to isomorphism of some squarefree m. That is, a(n) > 0 for n < 10000000.

It is conjectured that 0 never appears in this sequence.

			a(3)=609 since there are 3 groups of order 609 up to isomorphism, and 609 is the smallest squarefree integer such that there are 3 groups of that order.

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.

Cf. A046057 (m not necessarily squarefree).

A384185 Integers in A375491 in order of their first appearance.

Original entry on oeis.org

1, 2, 4, 6, 12, 5, 8, 24, 3, 7, 18, 16, 30, 36, 48, 10, 32, 14, 72, 9, 60, 96, 120, 19, 64, 13, 40, 144, 21, 35, 56, 38, 28, 90, 26, 240, 15, 192, 384, 44, 76, 360, 88, 80, 180, 168, 114, 54, 288, 112, 264, 25, 41, 33, 256, 98, 20, 55, 312, 128

Offset: 1

Robin Jones, May 21 2025

nonn

Does every positive integer appear in this sequence?

			a(1)=1 since the first squarefree integer (1) has 1 group of that order.
a(2)=2 since the first squarefree integer not to have a(1)=1 group of its order (6) has 2 groups of its order.
a(3)=4 since the first squarefree integer not to have a(1)=1 or a(2)=2 groups of its order (30) has 4 groups of its order.

Cf. A375491.

A383350 a(n) is the smallest integer k such that there are k+i groups of order a(n)+i, for i=1,...,n.

Original entry on oeis.org

0, 2, 72, 72, 2814120, 29436120

Offset: 1

Robin Jones, Apr 24 2025

The sequence is finite. For any multiple of 32, there are more than 32 groups of that order. Thus, the sequence 1,2,...,32 can't appear in A000001, and this sequence is at most 31 terms long.
The sequence is either 6 or 7 terms long. This can be shown by first showing every entry of A373650 is congruent to 24 mod 48. It then follows that if n is such that A000001(n+i) = i for i=1,2,3,4, then n+8 is a multiple of 16. But then A000001(n+8) >= 14, so we can't have A000001(n+i) = i for i=1,2,3,4,8.
From a(2) onwards, each entry is a multiple of 24, but not a multiple of 48.
a(7) > 223000000 if it exists.
Each entry is congruent to 0, 2 or 4 modulo 5.

			a(1) = 0 since there is 1 group of order 1.
a(2) = 2 since there is 1 group of order 3, 2 groups of order 4.

Cf. A373648, A373649, A373650, A381335.

A381335 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2,3,4,5.

Original entry on oeis.org

2814120, 22411272, 29436120, 46906920, 58734120, 59558520, 71510520, 106822200, 109673064, 117873720, 200250120, 213805272

Offset: 1

Robin Jones, Apr 19 2025

a(13) > 220000000 if it exists.
Each term is a multiple of 24. No terms are multiples of 48.
Each term is congruent to 0, 2 or 4 modulo 5. Terms can't be congruent to 5 modulo 7. I think they also can't be congruent to 3 modulo 7, but I haven't proven that yet.

			2814120 is in this sequence as there is 1 group of order 2814121 up to isomorphism, 2 of order 2814122, 3 of order 2814123, 4 of order 2814124, 5 of order 2814125.

Cf. A373648 (i=1,2), A373649 (i=1,2,3), A373650 (i=1,2,3,4).

A373649 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2,3.

Original entry on oeis.org

72, 864, 2064, 2172, 3972, 7932, 10332, 12632, 15120, 16536, 17472, 19236, 20316, 20336, 20664, 23772, 23880, 24420, 25092, 28920, 31476, 33132, 35136, 36876, 38172, 41016, 41772, 42060, 46020, 51480, 54084, 54392, 55596, 56196, 59700, 64512, 65820, 66600, 73272, 75972, 79020, 84744, 89784, 94980, 96672

Offset: 1

Robin Jones, Jun 12 2024

nonn

			72 is in this sequence as there is 1 group of order 73 up to isomorphism, 2 of order 74, 3 of order 75.

Robin Jones, Table of n, a(n) for n = 1..500

Cf. A373648 (i=1,2), A373650 (i=1,2,3,4), A381335 (i=1,2,3,4,5).

Equals A296024 - 1.

A373648 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2.

Original entry on oeis.org

2, 4, 12, 32, 36, 60, 72, 84, 132, 140, 144, 156, 176, 192, 212, 216, 276, 312, 344, 392, 396, 420, 444, 456, 480, 500, 536, 540, 552, 560, 564, 612, 660, 672, 696, 704, 716, 732, 744, 756, 792, 816, 864, 876, 884, 912, 932, 956, 972, 996, 1040, 1092, 1140, 1152, 1172, 1200

Offset: 1

Robin Jones, Jun 12 2024

nonn

All the terms are even. - Robin Jones, Apr 18 2025

			2 is a term since there is 1 group of order 3 up to isomorphism, 2 of order 4.

Robin Jones, Table of n, a(n) for n = 1..500

Equals A296023 - 1.
Cf. A373649 (i=1,2,3), A373650 (i=1,2,3,4), A381335 (i=1,2,3,4,5).
Subsequence of A003277 - 1.

cp's OEIS Frontend

User: Robin Jones

A384607 Possible values for the number of groups of order equal to a prime power, in order of size.

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A384606 Possible values for the number of groups of order equal to a prime power, in order of first appearance.

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A384609 Possible values for the number of nilpotent groups of a finite order, ordered by size.

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

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A384146 Smallest squarefree order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.

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A384185 Integers in A375491 in order of their first appearance.

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A383350 a(n) is the smallest integer k such that there are k+i groups of order a(n)+i, for i=1,...,n.

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A381335 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2,3,4,5.

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A373649 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2,3.

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A373648 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2.

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