cp's OEIS Frontend

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.

%I A053403 #25 Jun 22 2021 03:01:03
%S A053403 7,11,19,29,31,47,49,53,67,71,73,79,87,91,103,119,127,131,137,139,141,
%T A053403 142,143,146,147,151,155,179,191,193,201,203,211,213,219,223,227,229,
%U A053403 235,237,239,247,251,265,271,301,329,331,337,341,343,347,355,358,359
%N 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.
%C A053403 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.
%D A053403 R. Keith Dennis, The number of groups of order n, Cambridge Tracts in Mathematics, number 173.
%D A053403 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).
%H A053403 Charlie Neder, <a href="/A053403/b053403.txt">Table of n, a(n) for n = 1..508</a>
%H A053403 Claudia Spiro, <a href="http://cspiromathpapers.blogspot.fr/2011/">A Conjecture in Statistical Group theory</a>, Blog Entry, Dec 26 2011.
%H A053403 Claudia Spiro, <a href="/A053403/a053403.png">A Conjecture in Statistical Group theory</a>, Blog Entry, Dec 26 2011 [Cached copy, permission requested]
%H A053403 Claudia A. Spiro-Silverman, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6111.pdf">When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently</a>, Acta Arithmetica, 1992 | 61 | 1 | 1-12.
%H A053403 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%e A053403 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
%Y A053403 Cf. A000001, A046057.
%K A053403 nonn,nice
%O A053403 1,1
%A A053403 R. Keith Dennis (dennis(AT)math.cornell.edu), Jan 07 2000