|
| |
|
|
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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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 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),
|
|
|
LINKS
|
|
|
|
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
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
