Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Nov 04 2018 18:23:14
%S 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 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 29,39,11,29,41,13,29,43,17,27,45,25,21,47,33,15,49
%N 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 More formally:
%C - let S be the set of sequences of positive integers with positive indices,
%C - 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 v(n) = Sum_{i=1..w(n)} u(i + Sum_{j=1..n-1} w(j)),
%C or: Sum_{i=1..Sum_{j=1..n} w(j)} u(i) = Sum_{k=1..n} v(k),
%C (the sequence w gives the number of terms in each group)
%C - the set S with the binary relation R "u can be grouped and summed to produce v" is a partially ordered set,
%C - in particular, A028356 is R-related to A000027,
%C - for any u in S, A000012 is R-related to u (A000012 is the least element of S with respect to R),
%C - 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 - for any u, v and w in S, the function L satisfies:
%C L(u, u) = u,
%C L(u, v) = L(v, u),
%C L(u, L(v, w)) = L(L(u, v), w),
%C L(A000012, u) = A000012,
%C - this sequence corresponds to L(A000027, A000040).
%H Rémy Sigrist, <a href="/A321301/b321301.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A321301/a321301.gp.txt">PARI program for A321301</a>
%e The first terms of this sequence, alongside the groups summing to the first natural numbers and to the first prime numbers, are:
%e +-+---+-----+-------+---------+-----------+-------------+
%e - Natural numbers |1| 2 | 3 | 4 | 5 | 6 | 7 | ...
%e +-+-+-+---+-+-------+---------+---+-------+-------------+
%e - This sequence |1|1|1| 2 |1| 4 | 5 | 2 | 4 | 7 | ...
%e +-+-+-+---+-+-------+---------+---+-------+-------------+
%e - Prime numbers | 2 | 3 | 5 | 7 | 11 | ...
%e +---+-----+---------+-------------+---------------------+
%o (PARI) See Links section.
%Y Cf. A000012, A000027, A000040, A028356.
%K nonn,look
%O 1,4
%A _Rémy Sigrist_, Nov 03 2018