A373650 -id:A373650 - cp's OEIS frontend

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

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.

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.

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).

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.

cp's OEIS Frontend

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

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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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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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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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