A392698 Numbers whose reversed prime indices are the 0-prepended first sums of some finite sequence of distinct integers.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset

1,2

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

Also numbers k such that the trimmed 0-based partial alternating sums of the reversed prime indices of k are all distinct. 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. It is trimmed by removing the zeroth line (which is always k). For example, the trimmed 0-based partial alternating sums of q = (a,b,c,d,e) are:

pas(q,0)_1 = a

pas(q,0)_2 = -a + b

pas(q,0)_3 = a - b + c

pas(q,0)_4 = -a + b - c + d

pas(q,0)_5 = a - b + c - d + e

Conjecture: Integer partitions of this type (distinct trimmed 0-based partial alternating sums) appear to be counted by A000009. For example, the A000009(1) = 1 through A000009(9) = 8 partitions are:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(11) (21) (22) (32) (33) (43) (44) (54)

(31) (41) (42) (52) (53) (63)

(221) (51) (61) (62) (72)

(321) (331) (71) (81)

(421) (332) (432)

(431) (441)

(521) (531)

(621)

(3321)

Links

Table of n, a(n) for n=1..67.

Example

The reversed prime indices of 18 are (2,2,1), which is the 0-prepended first sums of (2,0,1), which are all distinct, so 18 is in the sequence.
The reversed prime indices of 75 are (3,3,2), which is the 0-prepended first sums of (3,0,2), which are all distinct, so 75 is in the sequence.
The reversed prime indices of 90 are (3,2,2,1), with 0-based partial alternating sums (0,3,-1,3,-2), trimmed (3,-1,3,-2), which are not all distinct, so 90 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  22: {1,5}

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], UnsameQ@@Rest[pas[Reverse[prix[#]], 0]]&]

Crossrefs

Partitions of this type appear to be counted by A000009, non-reversed A392704.

The non-trimmed version appears to be A005117, non-reversed A392695 (count A392701).

For constant instead of distinct we have A008578, non-reversed A392371.

Positions of strict rows in A391982 (sums A346697), non-reversed A391981 (sums A346699).

The non-reversed version is A392697.

For standard compositions we have A392709, counted by A392702.

Positions of strict trimmed rows in A392711 (sums A346697), non-reversed A392374 (sums A346699).

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).

A344610 counts partitions by sum and positive reverse-alternating sum.

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

A390673 ranks compositions with distinct first sums, count A390567.

A390989 counts partitions by sum of first sums, ranks A390990.

Cf. A000070, A001222, A316524, A344616, A345958, A346703, A390445, A390676, A392708, A392703, A392707.

Sequence in context: A166155 A381991 A394994 * A342525 A325457 A389213

Adjacent sequences: A392695 A392696 A392697 * A392699 A392700 A392701

Keyword

nonn

Author

Gus Wiseman, Jan 26 2026

Status

approved