A392697 Numbers whose prime indices are the 0-prepended first sums of some finite sequence of distinct integers.

1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 67, 68, 69, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

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 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

Links

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

Example

The prime indices of 45 are (2,2,3), with 0-based partial alternating sums (0,2,0,3), trimmed (2,0,3), which are distinct, so 45 is in the sequence.
The prime indices of 75 are (2,3,3), with 0-based partial alternating sums (0,2,1,2), trimmed (2,1,2), which are not distinct, so 75 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  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[prix[#], 0]]&]

Crossrefs

Positions of strict rows in A391981 (row sums A346699).

For constant instead of distinct we have A392371.

Non-trimmed strict positions in A392374 (row sums A346697).

The non-trimmed version is A392695, counted by A392701.

For reversed prime indices we have A392698, see A391982 (row sums A346697).

Partitions of this type are counted by A392704.

For trimmed standard compositions we have A392708 (counted by A392703).

For standard compositions we have A392709 (counted by A392702).

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, A000097, A001222, A005117, A316524, A344611, A344616, A345958, A346703, A390445, A390676, A391629.

Sequence in context: A039261 A039201 A039151 * A039110 A153350 A191919

Adjacent sequences: A392694 A392695 A392696 * A392698 A392699 A392700

Keyword

nonn

Author

Gus Wiseman, Jan 23 2026

Status

approved