A390429 Number of integer partitions that are the first sums of some integer partition of n > 0.

1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 95, 127, 167, 219, 284, 368, 471, 603, 765, 969, 1218, 1529, 1907, 2375, 2941, 3636, 4473, 5495, 6720, 8206, 9982, 12123, 14673, 17733, 21364, 25701, 30832, 36933, 44130, 52656, 62682, 74516, 88397, 104723, 123822

Offset

1,2

Comments

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=1..47.

Example

The partition (5,3,3,1) has first sums (8,6,4), so (8,6,4) is counted under a(12). For this conclusion we could also have used (4,4,2,2).
The partitions (3,3,1) and (4,2,2) both have first sums (6,4), so (6,4) is counted under both a(7) and a(8).
The a(1) = 1 through a(7) = 13 partitions:
  ()  ()   ()     ()       ()         ()           ()
      (2)  (3)    (4)      (5)        (6)          (7)
           (2,2)  (3,2)    (4,2)      (4,4)        (5,4)
                  (2,2,2)  (4,3)      (5,2)        (6,2)
                           (3,2,2)    (5,3)        (6,3)
                           (2,2,2,2)  (4,2,2)      (6,4)
                                      (4,3,2)      (4,4,3)
                                      (3,2,2,2)    (5,2,2)
                                      (2,2,2,2,2)  (5,3,2)
                                                   (4,2,2,2)
                                                   (4,3,2,2)
                                                   (3,2,2,2,2)
                                                   (2,2,2,2,2,2)

Mathematica

firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];
Table[Length[Union[firsums/@IntegerPartitions[n]]], {n, 15}]

Crossrefs

For subsets instead of partitions we appear to have A027934.

Partitions of this type are ranked by A390448.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A355523 counts distinct differences between prime indices, sums A110475.

A390307 gives first sums of prime indices.

Cf. A000041, A342527, A390362, A390446, A390447, A390449, A390567.

Sequence in context: A098889 A061481 A017824 * A343942 A094054 A001521

Adjacent sequences: A390426 A390427 A390428 * A390430 A390431 A390432

Keyword

nonn

Author

Gus Wiseman, Nov 13 2025

Status

approved