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.

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, 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, 29, 39, 11, 29, 41, 13, 29, 43, 17, 27, 45, 25, 21, 47, 33, 15, 49

Offset: 1

Author

Rémy Sigrist, Nov 03 2018

Keywords

• nonn
• look

Comments

More formally:

- let S be the set of sequences of positive integers with positive indices,

- 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:

v(n) = Sum_{i=1..w(n)} u(i + Sum_{j=1..n-1} w(j)),

or: Sum_{i=1..Sum_{j=1..n} w(j)} u(i) = Sum_{k=1..n} v(k),

(the sequence w gives the number of terms in each group)

- the set S with the binary relation R "u can be grouped and summed to produce v" is a partially ordered set,

- in particular, A028356 is R-related to A000027,

- for any u in S, A000012 is R-related to u (A000012 is the least element of S with respect to R),

- 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,

- for any u, v and w in S, the function L satisfies:

L(u, u) = u,

L(u, v) = L(v, u),

L(u, L(v, w)) = L(L(u, v), w),

L(A000012, u) = A000012,

- this sequence corresponds to L(A000027, A000040).

Examples

			The first terms of this sequence, alongside the groups summing to the first natural numbers and to the first prime numbers, are:
                  +-+---+-----+-------+---------+-----------+-------------+
- Natural numbers |1| 2 |  3  |   4   |    5    |     6     |      7      | ...
                  +-+-+-+---+-+-------+---------+---+-------+-------------+
- This sequence   |1|1|1| 2 |1|   4   |    5    | 2 |   4   |      7      | ...
                  +-+-+-+---+-+-------+---------+---+-------+-------------+
- Prime numbers   | 2 |  3  |    5    |      7      |          11         | ...
                  +---+-----+---------+-------------+---------------------+
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A321301

Crossrefs

Cf. A000012, A000027, A000040, A028356.

Programs

PARI

See Links section.