A325765 Number of integer partitions of n with a unique consecutive subsequence summing to every positive integer from 1 to n.

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3

Offset

0,4

Comments

After a(0) = 1, same as A032741(n + 1) (number of proper divisors of n + 1).

The Heinz numbers of these partitions are given by A325764.

Links

Table of n, a(n) for n=0..50.

Example

The a(1) = 1 through a(13) = 3 partitions:
  (1)  (11)  (21)   (1111)  (221)    (111111)  (2221)     (3311)
             (111)          (311)              (4111)     (11111111)
                            (11111)            (1111111)
.
  (22221)      (1111111111)  (33311)        (111111111111)  (2222221)
  (51111)                    (44111)                        (7111111)
  (111111111)                (222221)                       (1111111111111)
                             (611111)
                             (11111111111)

Mathematica

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n], normQ[Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]]&&UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 0, 20}]

Crossrefs

Cf. A000041, A002033, A103295, A103300, A143823, A169942, A325676, A325683, A325768, A325769, A325770.

Sequence in context: A294928 A325770 A305611 * A032741 A364818 A353338

Adjacent sequences: A325762 A325763 A325764 * A325766 A325767 A325768

Keyword

nonn,more

Author

Gus Wiseman, May 20 2019

Status

approved