A390307 Irregular triangle read by rows where row n lists the first sums of prime indices of n.

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

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

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[prix[n]], {n, 2, 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.

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

Heinz numbers for the prepended version are A390309, union A390364, complement A390365.

Reversing rows gives A390362.

For binary indices instead of prime indices we have A390431.

For standard compositions we have A390432.

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: A337121 A175470 A098534 * A317638 A390362 A002307

Adjacent sequences: A390304 A390305 A390306 * A390308 A390309 A390310

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 10 2025

Status

approved