A335106 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.

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

Offset

0,20

Comments

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).

Links

Table of n, a(n) for n=0..87.

David James Sycamore, Prime Partition Trees

Example

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;
...

Mathematica

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 *)

Crossrefs

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

Keyword

nonn,easy,tabf

Author

David James Sycamore, Michael De Vlieger, May 23 2020

Extensions

More terms from Giovanni Resta, May 31 2020

Status

approved