cp's OEIS Frontend

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.

%I A335106 #30 Aug 02 2020 15:40:33
%S A335106 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,
%T A335106 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,
%U A335106 5,3,2,0,3,6,7,3,2,1,1,1,3,7,7,4,3,1,0,4
%N 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.
%C A335106 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).
%H A335106 David James Sycamore, <a href="/A335106/a335106.jpg">Prime Partition Trees</a>
%e A335106 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:
%e A335106 0;
%e A335106 0;
%e A335106 1;
%e A335106 0, 1;
%e A335106 1;
%e A335106 0, 1, 1;
%e A335106 1, 1;
%e A335106 0, 1, 1, 1;
%e A335106 1, 1, 1;
%e A335106 0, 2, 1, 1;
%e A335106 1, 1, 2, 1;
%e A335106 0, 2, 2, 1, 1;
%e A335106 1, 2, 2, 2;
%e A335106 0, 2, 3, 2, 1, 1;
%e A335106 1, 2, 3, 3, 1;
%e A335106 0, 3, 4, 3, 1, 1;
%e A335106 1, 2, 4, 4, 2, 1;
%e A335106 0, 3, 5, 5, 2, 1, 1;
%e A335106 ...
%t A335106 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 *)
%t A335106 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 *)
%Y A335106 Cf. A000040, A000607, A000720, A333365, A331634, A335285.
%Y A335106 Row sums gives A000607 for n > 1. Length of n-th row is A000720(A335285(n)) for n >1.
%Y A335106 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).
%K A335106 nonn,easy,tabf
%O A335106 0,20
%A A335106 _David James Sycamore_, _Michael De Vlieger_, May 23 2020
%E A335106 More terms from _Giovanni Resta_, May 31 2020