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A321301 Lexicographically last sequence of positive integers whose terms can be grouped and summed to produce the natural numbers as well as the prime numbers.

%I A321301 #16 Nov 04 2018 18:23:14
%S A321301 1,1,1,2,1,4,5,2,4,7,8,5,4,10,3,8,11,1,13,9,5,15,9,7,17,7,11,19,7,13,
%T A321301 21,7,15,23,5,19,25,3,23,27,3,25,29,5,25,31,5,27,33,7,27,35,9,27,37,9,
%U A321301 29,39,11,29,41,13,29,43,17,27,45,25,21,47,33,15,49
%N A321301 Lexicographically last sequence of positive integers whose terms can be grouped and summed to produce the natural numbers as well as the prime numbers.
%C A321301 More formally:
%C A321301 - let S be the set of sequences of positive integers with positive indices,
%C A321301 - for any u and v in S, the terms of u can be grouped and summed to produce v iff there is an element w in S such that for any n > 0:
%C A321301       v(n) = Sum_{i=1..w(n)} u(i + Sum_{j=1..n-1} w(j)),
%C A321301       or: Sum_{i=1..Sum_{j=1..n} w(j)} u(i) = Sum_{k=1..n} v(k),
%C A321301       (the sequence w gives the number of terms in each group)
%C A321301 - the set S with the binary relation R "u can be grouped and summed to produce v" is a partially ordered set,
%C A321301 - in particular, A028356 is R-related to A000027,
%C A321301 - for any u in S, A000012 is R-related to u (A000012 is the least element of S with respect to R),
%C A321301 - for any u and v, let L(u, v) denote the lexicographically last element of S that is R-related both to u and to v,
%C A321301 - for any u, v and w in S, the function L satisfies:
%C A321301        L(u, u) = u,
%C A321301        L(u, v) = L(v, u),
%C A321301        L(u, L(v, w)) = L(L(u, v), w),
%C A321301        L(A000012, u) = A000012,
%C A321301 - this sequence corresponds to L(A000027, A000040).
%H A321301 Rémy Sigrist, <a href="/A321301/b321301.txt">Table of n, a(n) for n = 1..10000</a>
%H A321301 Rémy Sigrist, <a href="/A321301/a321301.gp.txt">PARI program for A321301</a>
%e A321301 The first terms of this sequence, alongside the groups summing to the first natural numbers and to the first prime numbers, are:
%e A321301                   +-+---+-----+-------+---------+-----------+-------------+
%e A321301 - Natural numbers |1| 2 |  3  |   4   |    5    |     6     |      7      | ...
%e A321301                   +-+-+-+---+-+-------+---------+---+-------+-------------+
%e A321301 - This sequence   |1|1|1| 2 |1|   4   |    5    | 2 |   4   |      7      | ...
%e A321301                   +-+-+-+---+-+-------+---------+---+-------+-------------+
%e A321301 - Prime numbers   | 2 |  3  |    5    |      7      |          11         | ...
%e A321301                   +---+-----+---------+-------------+---------------------+
%o A321301 (PARI) See Links section.
%Y A321301 Cf. A000012, A000027, A000040, A028356.
%K A321301 nonn,look
%O A321301 1,4
%A A321301 _Rémy Sigrist_, Nov 03 2018