%I #7 May 03 2019 08:37:21

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,30,

%T 31,32,33,34,35,37,38,39,41,43,46,47,49,50,51,53,54,55,57,58,59,61,62,

%U 64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87,89

%N Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.

%C First differs from A325361 in lacking 150.

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

%C 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).

%C 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.

%C The enumeration of these partitions by sum is given by A325353.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:

%e 12: {1,1,2}

%e 20: {1,1,3}

%e 24: {1,1,1,2}

%e 28: {1,1,4}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 42: {1,2,4}

%e 44: {1,1,5}

%e 45: {2,2,3}

%e 48: {1,1,1,1,2}

%e 52: {1,1,6}

%e 56: {1,1,1,4}

%e 60: {1,1,2,3}

%e 63: {2,2,4}

%e 66: {1,2,5}

%e 68: {1,1,7}

%e 72: {1,1,1,2,2}

%e 76: {1,1,8}

%e 78: {1,2,6}

%e 80: {1,1,1,1,3}

%e The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,-1,-1) and (-1,0) respectively.

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[100],And@@Table[GreaterEqual@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

%Y Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019