A373650 Integers k such that there are i groups of order k+i up to isomorphism, for i=1,2,3,4.

72, 20664, 66600, 84744, 89784, 141240, 175032, 232680, 271272, 288072, 378984, 428472, 620472, 697320, 740520, 789672, 792360, 1016472, 1063272, 1207704, 1250472, 1304472, 1338600, 1570584, 1617672, 1628472, 1844472, 2150712, 2186472, 2283672, 2399112, 2427672

Offset: 1

Author

Robin Jones, Jun 12 2024

Keywords

• nonn

Comments

Comment from Robin Jones, May 05 2025: (Start)

Each term is a multiple of 24. No terms are multiples of 48. That is, each term is congruent to 24 mod 48. Equivalently, 8 divides a(n), 3 divides a(n), but 16 does not divide a(n), for all n.

Each term is congruent to 0, 2 or 4 modulo 5. Terms can't be congruent to 5 modulo 7.

Conjectures (verified to 783 terms):

Terms are never 3 mod 7.

Terms are never 5 or 9 mod 11.

Terms are never 9 or 11 mod 13.

Terms are never 13 or 15 mod 17.

Terms are never 17 mod 19.

Terms are never 19 or 21 mod 23.

In fact it looks like it could be true that for any prime p > 3, terms are never congruent to p-2 mod p.

If n = p-2 mod p, then n+2 = 2pm for some m, since n is even. Then if m > 1, one can show that this will always have 4 or more groups. Thus m=1 and n=2p-2. One must show that n can never be of the form 2p-2 if this conjecture is true.

It looks like the terms are also unevenly distributed modulo 5, 7, 11, and so on, within the classes that terms can belong to. For example, modulo 5, it seems like terms are most commonly congruent to 2 modulo 5, and it is relatively rare for terms to be congruent to 4 modulo 5 (see plots in Links).

(End)

Examples

			72 is in this sequence as there is 1 group of order 73 up to isomorphism, 2 of order 74, 3 of order 75, 4 of order 76.
		

Links

• David Radcliffe, Table of n, a(n) for n = 1..6405 (terms 1..1418 and 1420..5512 from Robin Jones).
• Robin Jones, Graph of n, a(n) mod 5 for n = 1..783
• Robin Jones, Graph of n, a(n) mod 7 for n = 1..783
• Robin Jones, Graph of n, a(n) mod 11 for n = 1..783

Crossrefs

Cf. A373648 (i=1,2), A373649 (i=1,2,3), A381335 (i=1,2,3,4,5).

Programs

Magma

for x in [1 .. 100000] do //get the terms up to 100000
    if NumberOfSmallGroups(x+1) eq 1 then
        if NumberOfSmallGroups(x+2) eq 2 then
            if NumberOfSmallGroups(x+3) eq 3 then
                if NumberOfSmallGroups(x+4) eq 4 then
                    x;
                end if;
            end if;
        end if;
    end if;
end for; // Robin Jones, Apr 18 2025