A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

1, 2, 3, 4, 5, 7, 9, 10, 11, 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, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325324.

Links

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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[200], UnsameQ@@Differences[Append[primeptn[#], 0]]&]

Crossrefs

Positions of squarefree numbers in A325390.

Cf. A056239, A112798, A130091, A320348, A325324, A325327, A325362, A325364, A325366, A325368, A325388, A325405, A325407, A325460, A325461.

Sequence in context: A047363 A191917 A228894 * A160718 A122090 A066050

Adjacent sequences: A325364 A325365 A325366 * A325368 A325369 A325370

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved