A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74

Offset: 1

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

Comments

First differs from A109427 in lacking 54.

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

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).

The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.

The enumeration of these partitions by sum is given by A325354.

Examples

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  147: {2,4,4}
  150: {1,2,3,3}
  154: {1,4,5}
  162: {1,2,2,2,2}
		

Links

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

Crossrefs

Cf. A007294, A056239, A112798, A240026, A325354, A325360, A325362, A325394, A325397, A325398, A325399, A325405, A325467.

Programs

Mathematica

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]