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

2814120, 22411272, 29436120, 39276120, 46906920, 58734120, 59558520, 71510520, 106822200, 109673064, 117873720, 200250120, 213805272, 222954120, 257287944, 264020472, 287289720, 324510120, 329876472, 335421912, 337416120, 346959192, 371214120, 379233192, 385413672, 449712120, 452121672

Offset

1,1

Comments

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.

A necessary condition for k to be in this sequence is k = 48*m + 24 where 48*m + 25 is squarefree and 24*m + 13 and 12*m + 7 are prime. If k meets this condition, then it is sufficient to check k+3 has 3 groups, k+5 has 5 groups, and k+1 has 1 group. - Gabriel Eiseman, Jan 02 2026

Links

Gabriel Eiseman, Table of n, a(n) for n = 1..10000

Keith Conrad, Groups of order p^3.

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.

G. A. Miller, Orders for which There Exist Exactly Four or Five Groups, Proceedings of the National Academy of Sciences of the United States of America, vol. 18, no. 7, 1932, pp. 511-514.

Jørn B. Olsson, Three-group numbers, 2006.

Example

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.

Crossrefs

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

Sequence in context: A234189 A321056 A033534 * A204744 A258563 A204413

Adjacent sequences: A381332 A381333 A381334 * A381336 A381337 A381338

Keyword

nonn

Author

Robin Jones, Apr 19 2025

Extensions

Missing 39276120 inserted and more terms from Gabriel Eiseman, Jan 02 2026

Status

approved