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A335106
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Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.
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4
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0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 0, 2, 3, 2, 1, 1, 1, 2, 3, 3, 1, 0, 3, 4, 3, 1, 1, 1, 2, 4, 4, 2, 1, 0, 3, 5, 5, 2, 1, 1, 1, 3, 5, 5, 3, 2, 0, 3, 6, 7, 3, 2, 1, 1, 1, 3, 7, 7, 4, 3, 1, 0, 4
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OFFSET
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0,20
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COMMENTS
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Let L(n) be the set of least part primes of all prime partitions of n, and let G(n) be corresponding set of greatest prime parts. All prime partitions, commencing with p in L(n) and terminating with q in G(n) can be shown as unique pathways on a partition tree of n; see link for details. |G(n)| = A000607(n).
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LINKS
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Table of n, a(n) for n=0..87.
David James Sycamore, Prime Partition Trees
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EXAMPLE
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A000607(10) = 5 and the prime partitions of 10 are: (2,2,2,2,2), (2,2,3,3), (2,3,5), (5,5) and (3,7). Thus G(10) = {2,3,5,5,7}, and consequently row 10 is [1,1,2,1]. In the table below, for n >= 2, 0 is used to indicate when prime(k) is not in G(n) and is less than the greatest member of G(n), otherwise the entry for prime(k) not in G(n) is left empty. For n >= 2 the sum of entries in the n-th row is |G(n)| = A000607(n). Triangle T(n,k) begins:
0;
0;
1;
0, 1;
1;
0, 1, 1;
1, 1;
0, 1, 1, 1;
1, 1, 1;
0, 2, 1, 1;
1, 1, 2, 1;
0, 2, 2, 1, 1;
1, 2, 2, 2;
0, 2, 3, 2, 1, 1;
1, 2, 3, 3, 1;
0, 3, 4, 3, 1, 1;
1, 2, 4, 4, 2, 1;
0, 3, 5, 5, 2, 1, 1;
...
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MATHEMATICA
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Flatten@ Block[{nn = 22, t}, t = Block[{s = {Prime@ PrimePi@ nn}}, KeySort@ Merge[#, Identity] &@ Join[{0 -> {}, 1 -> {}}, Reap[Do[If[# <= nn, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1] ] ] &@Total[s], {i, Infinity}]][[-1, -1]] ] ]; Array[Function[p, If[! IntegerQ@ First@ p, {0}, Array[Count[p, Prime@ #] &, PrimePi@ Max@ p]]]@ Map[Max, t[[#]]] &, Max@ Keys@ t]] (* Michael De Vlieger, May 23 2020 *)
row[0]={0}; row[k_] := Join[If[OddQ@k, {0}, {}], Last /@ Tally@ Sort[ First /@ IntegerPartitions[k, All, Prime@ Range@ PrimePi@ k]]]; Join @@ Array[row, 20, 0] (* Giovanni Resta, May 31 2020 *)
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CROSSREFS
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Cf. A000040, A000607, A000720, A333365, A331634, A335285.
Row sums gives A000607 for n > 1. Length of n-th row is A000720(A335285(n)) for n >1.
Number of partition of n in the first k primes: A059841 (k = 1), A103221 (k = 2), A025795 (k = 3), A029144 (k = 4), A140952 (k = 5), A140953 (k = 6).
Sequence in context: A016024 A238988 A261013 * A093518 A128184 A025450
Adjacent sequences: A335103 A335104 A335105 * A335107 A335108 A335109
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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David James Sycamore, Michael De Vlieger, May 23 2020
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EXTENSIONS
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More terms from Giovanni Resta, May 31 2020
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STATUS
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approved
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