%I #6 Oct 31 2022 22:05:44
%S 1,2,1,2,3,1,3,4,1,2,3,2,4,1,4,5,1,2,4,6,1,5,2,5,1,2,3,4,7,1,3,5,8,1,
%T 2,5,2,6,1,6,9,1,2,3,5,3,6,1,7,2,4,6,1,2,6,10,1,3,6,11,1,2,3,4,5,2,7,
%U 1,8,3,7,1,2,4,6,12,1,9,2,8,1,2,3,6,13
%N Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%e Triangle begins:
%e 2: 1
%e 3: 2
%e 4: 1 2
%e 5: 3
%e 6: 1 3
%e 7: 4
%e 8: 1 2 3
%e 9: 2 4
%e 10: 1 4
%e 11: 5
%e 12: 1 2 4
%e 13: 6
%e 14: 1 5
%e 15: 2 5
%e 16: 1 2 3 4
%e 17: 7
%e 18: 1 3 5
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Accumulate[primeMS[n]],{n,30}]
%Y Row-lengths are A001222.
%Y First element in each row is A055396.
%Y Last element in each row is A056239.
%Y Rows are the partial sums of rows of A112798.
%Y Row-sums are A318283.
%Y Sorted Heinz numbers of the rows are A325362.
%Y The version for standard compositions is A358134.
%Y Rows are ranked by A358137.
%Y A000041 counts partitions, strict A000009.
%Y A003963 multiplies prime indices.
%Y A056239 adds up prime indices.
%Y Cf. A000720, A001221, A355536, A358133.
%K nonn,tabf
%O 2,2
%A _Gus Wiseman_, Oct 31 2022