A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 3, 1, 1, 3, 2, 5, 1, 9, 2, 1, 1, 1, 1, 3, 6, 1, 1, 1, 2, 4, 1, 1, 10, 2, 2, 1, 11, 1, 1, 1, 1, 1, 4, 2, 7, 1, 2, 3, 1, 2, 1, 1, 12, 8, 1, 5, 2, 3, 1, 1, 1

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.

The augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)_i = q_i - q_{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

Links

Table of n, a(n) for n=2..85.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 2 1
   7: 4
   8: 1 1 1
   9: 1 2
  10: 3 1
  11: 5
  12: 2 1 1
  13: 6
  14: 4 1
  15: 2 2
  16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[aug[Reverse[primeMS[n]]], {n, 30}]

Crossrefs

Crossrefs found in the link are not repeated here.

Row-lengths are A001222.

Row-sums are A252464

Other similar triangles are A287352, A091602.

Constant rows have indices A307824.

The Heinz numbers of the rows are A325351.

Strict rows have indices A325366.

Row minima are A355531, non-augmented A355524, also A355525.

Row maxima are A355535, non-augmented A286470, also A355526.

The non-augmented version is A355536, also A355533.

A112798 lists prime indices, sum A056239.

Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

Sequence in context: A275676 A025831 A184751 * A296150 A079673 A124829

Adjacent sequences: A355531 A355532 A355533 * A355535 A355536 A355537

Keyword

nonn,tabf

Author

Gus Wiseman, Jul 12 2022

Status

approved