A390362 Irregular triangle read by rows where row n lists the first sums of reversed prime indices of n.

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

Offset

4,1

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 first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Links

Table of n, a(n) for n=4..88.

Example

The reversed prime indices of 18 are (2,2,1), with first sums (4,3), so row 18 is (4,3).
Triangle begins:
    2:
    3:
    4:  2
    5:
    6:  3
    7:
    8:  2  2
    9:  4
   10:  4
   11:
   12:  3  2
   13:
   14:  5
   15:  5
   16:  2  2  2
   17:
   18:  4  3
   19:
   20:  4  2
   21:  6
   22:  6
   23:
   24:  3  2  2

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];
Table[firsums[Reverse[prix[n]]], {n, 100}]

Crossrefs

Positions of strict rows are A004709.

Row-lengths are A252736 = A001222 minus one.

Positions of constant rows are A354144.

For first differences instead of first sums we have A355536 with rows reversed.

Reversing rows gives A390307.

The prepended reverse is A390308, for differences A358169, reverse A355534.

The prepended Heinz numbers are A390309, union A390364, complement A390365.

For standard compositions we have A390432 with rows reversed.

The Heinz numbers of the rows are A390449, sorted A390448.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A342527 counts compositions with all equal first sums.

A390567 counts compositions with all distinct first sums.

Cf. A055396, A093641, A174090, A243055, A243056, A286470, A325328, A355523, A390363.

Sequence in context: A098534 A390307 A317638 * A002307 A287707 A029247

Adjacent sequences: A390359 A390360 A390361 * A390363 A390364 A390365

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 11 2025

Status

approved