1,2

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, ...).

Also numbers whose k-based partial alternating sums of prime indices are weakly increasing and > 1 for some k. Here, the k-based partial alternating sums of a finite sequence q are given by pas(q,k)_j = (-1)^j * k + Sum_{i=1..j} (-1)^(i+j) * q_i. This is a signed version of the partial sums transformation, inverse to the "first sums" transformation. For example, the k-based partial alternating sums of q = (a,b,c,d,e) are:

pas(q,k)_0 = k

pas(q,k)_1 = -k + a

pas(q,k)_2 = k - a + b

pas(q,k)_3 = -k + a - b + c

pas(q,k)_4 = k - a + b - c + d

pas(q,k)_5 = -k + a - b + c - d + e

The prime indices of 77 are (4,5), which is the first sums of (2,2,3), so 77 is in the sequence.

The prime indices of 121 are (5,5), with k-based partial alternating sums (k,5-k,k), which are not weakly increasing and > 1 for any choice of k, so 121 is not in the sequence.

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

pas[y_, k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]], {i, j}], {j, 0, Length[y]}];

Select[Range[100], #==1||Select[Table[pas[prix[#], k], {k, 2, Max[prix[#]]}], LessEqual@@#&&Min@@#>1&]!={}&]

For postpositive compositions instead of partitions we appear to have A007775.

Allowing all parts >= 1 gives A390448, counted by A390446.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum (reverse A344612).

A344606 counts alternating permutations of prime indices.

A390307 lists first sums of prime indices, see A390362, A390449.

A392374 lists 0-based partial alternating sums of prime indices, row sums A346699.

Cf. A001222, A316524, A344616, A345958, A390429, A390431, A390445, A390990, A391981, A391982, A392695.

Sequence in context: A161850 A007775 A070884 * A135777 A090459 A090417

Adjacent sequences: A392707 A392708 A392709 * A392711 A392712 A392716

Gus Wiseman, Feb 07 2026

approved