OFFSET
1,4
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,
- 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),
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A321301
EXAMPLE
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 | ...
+---+-----+---------+-------------+---------------------+
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Rémy Sigrist, Nov 03 2018
STATUS
approved