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

1,1

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.

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

Example

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

Crossrefs

Cf. A000001, A046057.

Sequence in context: A370010 A141337 A192187 * A032672 A235806 A238501

Adjacent sequences: A053400 A053401 A053402 * A053404 A053405 A053406

Keyword

nonn,nice

Author

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

Status

approved