A053403 Consider the set P of pairs (a,b) generated by the rules: (1,1) is in P; if (a,b) is in P then (b,a+b) is in P; if (a,b) and (a',b') are in P then (aa', bb') is in P. Sequence gives numbers not appearing in P.

7, 11, 19, 29, 31, 47, 49, 53, 67, 71, 73, 79, 87, 91, 103, 119, 127, 131, 137, 139, 141, 142, 143, 146, 147, 151, 155, 179, 191, 193, 201, 203, 211, 213, 219, 223, 227, 229, 235, 237, 239, 247, 251, 265, 271, 301, 329, 331, 337, 341, 343, 347, 355, 358, 359

Offset: 1

Author

R. Keith Dennis (dennis(AT)math.cornell.edu), Jan 07 2000

Keywords

• nonn
• nice

Comments

Sequence has 508 known terms, the largest of which is 55487. Conjecturally it is finite. If it is and 55487 is the largest term, then the function the number of groups of order n takes on all positive integers as values.

Examples

			The pairs with b <= 7 are (1,1), (1,2), (1,4), (2,3), (2,6), (3,5), and (4,5).  Since none of these has b = 7, 7 can never appear in P. - _Charlie Neder_, Feb 01 2019
		

References

• R. Keith Dennis, The number of groups of order n, Cambridge Tracts in Mathematics, number 173.
• Claudia A. Spiro, Local distribution results for the group-counting function at positive integers. In Proceedings of the Sundance conference on combinatorics and related topics (Sundance, Utah, 1985). Congr. Numer. 50 (1985), 107-110. MR0833542 (87g:11117).

Links

• Charlie Neder, Table of n, a(n) for n = 1..508
• Claudia Spiro, A Conjecture in Statistical Group theory, Blog Entry, Dec 26 2011.
• Claudia Spiro, A Conjecture in Statistical Group theory, Blog Entry, Dec 26 2011 [Cached copy, permission requested]
• Claudia A. Spiro-Silverman, When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently, Acta Arithmetica, 1992 | 61 | 1 | 1-12.
• Index entries for sequences related to groups

Crossrefs

Cf. A000001, A046057.