A392707 Numbers whose prime indices have strictly increasing 0-based partial alternating sums.

1, 2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 110, 111, 113, 115, 118, 122, 123, 127, 129, 130

Offset

1,2

Comments

First differs from A325460 in having 110.

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 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.

Also numbers whose prime indices are the first sums of some finite strictly increasing sequence of nonnegative integers beginning with 0, where the first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Conjecture: Reversed integer partitions of this type (with strictly increasing 0-based partial alternating sums) are counted by A053251.

Links

Table of n, a(n) for n=1..63.

Example

The prime indices of 10 are (1,3), with 0-based partial alternating sums (0,1,2), which are strictly increasing, so 10 is in the sequence.
The prime indices of 130 are (1,3,6), with 0-based partial alternating sums (0,1,2,4), which are strictly increasing, so 130 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  26: {1,6}
Sequences of this type grouped by sum begin:
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)      (10)
                 (1,3)  (1,4)  (1,5)  (1,6)  (1,7)  (1,8)    (1,9)
                                      (2,5)  (2,6)  (2,7)    (2,8)
                                                    (1,3,5)  (3,7)
                                                             (1,3,6)

Mathematica

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], Less@@pas[prix[#], 0]&]

Crossrefs

Partitions of this type appear to be counted by A053251.

The first term with omega > k is A066205(k).

These are positions of strictly increasing rows in A391981, sums A346699.

For weakly instead of strictly increasing we have A392694, apparent count A000009.

For distinct instead of strictly increasing we have A392695, counted by A392701.

For distinct and trimmed we have A392697, counted by A392704.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A344606 counts alternating permutations of prime indices.

A390307 lists first sums of prime indices, see A390362, A390448, A390449.

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

A392708 ranks comps with distinct 0-based partial alternating sums, count A392703.

Cf. A001222, A005117, A316524, A344616, A345958, A346697, A390445, A390446, A390990, A391982, A392371, A392711.

Sequence in context: A325160 A258613 A395643 * A325460 A002269 A327445

Adjacent sequences: A392704 A392705 A392706 * A392708 A392709 A392710

Keyword

nonn

Author

Gus Wiseman, Feb 05 2026

Status

approved