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Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).
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%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