A358136 Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).

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, 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, 1, 8, 3, 7, 1, 2, 4, 6, 12, 1, 9, 2, 8, 1, 2, 3, 6, 13

Offset

2,2

Comments

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.

Links

Table of n, a(n) for n=2..86.

Example

Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 1 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 1 4
  11: 5
  12: 1 2 4
  13: 6
  14: 1 5
  15: 2 5
  16: 1 2 3 4
  17: 7
  18: 1 3 5

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Accumulate[primeMS[n]], {n, 30}]

Crossrefs

Row-lengths are A001222.

First element in each row is A055396.

Last element in each row is A056239.

Rows are the partial sums of rows of A112798.

Row-sums are A318283.

Sorted Heinz numbers of the rows are A325362.

The version for standard compositions is A358134.

Rows are ranked by A358137.

A000041 counts partitions, strict A000009.

A003963 multiplies prime indices.

A056239 adds up prime indices.

Cf. A000720, A001221, A355536, A358133.

Sequence in context: A225597 A116908 A265146 * A325537 A072851 A246688

Adjacent sequences: A358133 A358134 A358135 * A358137 A358138 A358139

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 31 2022

Status

approved