A325360 - OEIS

%I #4 May 02 2019 08:55:06

%S 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,

%T 28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,

%U 53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,71,73

%N Heinz numbers of integer partitions whose differences are weakly increasing.

%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 (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (-3,-2).

%C The enumeration of these partitions by sum is given by A240026.

%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    18: {1,2,2}

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

%e    50: {1,3,3}

%e    54: {1,2,2,2}

%e    70: {1,3,4}

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

%e    75: {2,3,3}

%e    90: {1,2,2,3}

%e    98: {1,4,4}

%e   100: {1,1,3,3}

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

%t Select[Range[100],OrderedQ[Differences[primeptn[#]]]&]

%Y Cf. A007294, A056239, A112798, A240026, A325328, A325352, A325354, A325360, A325361, A325368, A325394, A325400.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019