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A132075 A conjectured permutation of the positive integers such that for every n, a(n) is the largest number among a(1), a(2), ..., a(n) that when added to a(n+1) gives a prime.

%I A132075 #28 Nov 19 2021 03:24:59
%S A132075 1,2,3,4,7,6,5,14,9,10,13,16,15,8,11,20,17,12,19,24,23,18,25,22,21,26,
%T A132075 27,34,33,28,31,30,29,32,35,36,37,46,43,40,39,44,45,38,41,42,47,50,59,
%U A132075 54,55,58,51,62,65,48,61,52,57,56,53,60,49,64,63,68,69,70,67,72,77,80,71,66,73,78,79,84,83,90,89,74,75,76,81,82,85,88,93,86,95,104,107,92
%N A132075 A conjectured permutation of the positive integers such that for every n, a(n) is the largest number among a(1), a(2), ..., a(n) that when added to a(n+1) gives a prime.
%C A132075 The terms are defined as follows. Start by choosing the initial terms: 1, 2, 3. Then write the rows of table A088643 backwards but always leave off the last three quarters of the terms. This gives: [], [], [], [1], [1], [1], [1], [1, 2], [1, 2], [1, 2], [1, 4], [1, 4, 3,], [1, 4, 3] etc. Then build the sequence up by repeatedly choosing the first such truncated row that extends the terms already chosen. [Edited by _Peter Munn_, Aug 19 2021]
%C A132075 It is not until the 26th truncated row - [1, 2, 3, 4, 7, 6] - that the initial list is extended at all. It is unclear whether this process can be continued indefinitely, although I have verified by computer that it generates a sequence of at least 2000 terms. Conjecturally: (1) the sequence is infinite, (2) it is the unique sequence containing infinitely many complete rows of table A088643, and (3) for every n > 0 there exists N > 0 such that the first n terms of this sequence are contained in every row of table A088643 from the N-th onwards.
%C A132075 Maybe the idea could be expressed more concisely by defining this sequence as the limit of the reversed rows of A088643? - _M. F. Hasler_, Aug 04 2021
%C A132075 It seems we do not know of an existence proof for the limit of the reversed rows of A088643. - _Peter Munn_, Aug 19 2021
%H A132075 Peter Munn, <a href="/A255312/a255312_1.txt">Illustration of the relationship between A088643, this sequence and A255312</a>.
%Y A132075 Cf. A088643.
%K A132075 easy,nonn
%O A132075 1,2
%A A132075 _Paul Boddington_, Oct 30 2007, Mar 06 2010
%E A132075 Name edited by _Peter Munn_, Aug 19 2021