A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset

1,2

Comments

First differs from A299702 in having 462.

The enumeration of these partitions by sum is given by A325769.

Links

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

Example

Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], UnsameQ@@Total/@Union[ReplaceList[primeMS[#], {___, s__, ___}:>{s}]]&]

Crossrefs

Complement of A325777.

Cf. A002033, A056239, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325779.

Sequence in context: A085235 A364461 A160453 * A364531 A299702 A348577

Adjacent sequences: A325775 A325776 A325777 * A325779 A325780 A325781

Keyword

nonn

Author

Gus Wiseman, May 20 2019

Status

approved