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%I #4 May 02 2019 08:55:13
%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 differences are weakly decreasing.
%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, 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 A320466.
%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}
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],GreaterEqual@@Differences[primeptn[#]]&]
%Y Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 02 2019
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