A356958 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (b-a+1, ..., y-a+1, z-a+1).

1, 2, 1, 1, 1, 3, 1, 2, 4, 2, 1, 1, 1, 2, 2, 1, 3, 3, 5, 1, 1, 2, 1, 6, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 7, 2, 1, 2, 2, 8, 5, 1, 1, 3, 2, 4, 1, 5, 1, 2, 9, 1, 1, 1, 2, 1, 3, 3, 6, 1, 6, 2, 2, 2, 3, 1, 1, 4, 7, 10, 1, 2, 3, 11, 1, 3, 1, 1, 1, 1, 1, 4, 2, 5

Offset

1,2

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=1..85.

Example

Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  1 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
For example, the prime indices of 315 are (2,2,3,4), so row 315 is (2,3,4) - 2 + 1 = (1,2,3).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, {}, 1-First[primeMS[n]]+Rest[primeMS[n]]], {n, 100}]

Crossrefs

Row lengths are A001222(n) - 1.

Indices of empty rows are A008578.

Even bisection is A112798.

Heinz numbers of rows are A246277.

An opposite version is A358172, Heinz numbers A358195.

Row sums are A359358(n) + A001222(n) - 1.

A112798 list prime indices, sum A056239.

A243055 subtracts the least prime index from the greatest.

Cf. A055396, A124010, A241916, A253565, A325351, A325352, A326844, A355534, A355536, A358137.

Sequence in context: A365466 A095136 A105540 * A057043 A325307 A211095

Adjacent sequences: A356955 A356956 A356957 * A356959 A356960 A356961

Keyword

nonn,tabf

Author

Gus Wiseman, Dec 27 2022

Status

approved