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A321301
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Lexicographically last sequence of positive integers whose terms can be grouped and summed to produce the natural numbers as well as the prime numbers.
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1
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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
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listen;
history;
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OFFSET
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1,4
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COMMENTS
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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),
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LINKS
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EXAMPLE
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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 | ...
+---+-----+---------+-------------+---------------------+
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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