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A281013
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Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.
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16
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1, 2, 2, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 1, 4, 2, 5, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 3, 3, 1, 4, 1, 1, 1, 4, 1, 2, 4, 2, 1, 4, 3, 5, 1, 1, 5, 2, 6, 1, 7
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OFFSET
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1,2
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COMMENTS
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The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details.
These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019
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LINKS
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FORMULA
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Row lengths are A059966(n) = number of prime compositions of n.
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EXAMPLE
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The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are:
(1) = (1)
(12) = (1)*(2)
(122) = (1)*(2)*(2)
(1221) = (1)*(221)
(12211) = (1)*(2211)
(122112) = (1)*(2)*(2211)
(1221121) = (1)*(221121)
(12211212) = (1)*(2)*(221121)
(122112122) = (1)*(2)*(2)*(221121)
(1221121221) = (1)*(221)*(221121)
(12211212212) = (1)*(2)*(221)*(221121)
(122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1 2 2 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 1 1
3 3 1 2 2 1 2 2 1 1 2 1 2 1 1
4 3 1 1 3 1 1 1 2 2 1 1 1
3 2 3 1 2 2 2 2 1
4 1 3 2 1 3 1 1 1 1
5 4 1 1 3 1 1 2
4 2 3 1 2 1
5 1 3 2 1 1
6 3 2 2
3 3 1
4 1 1 1
4 1 2
4 2 1
4 3
5 1 1
5 2
6 1
7.
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MATHEMATICA
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colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n], colynQ], lexsort], {n, 5}] (* Gus Wiseman, Nov 15 2019 *)
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CROSSREFS
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The binary non-"co" version is A102659.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is co-Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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