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A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.

%I A325361 #4 May 02 2019 08:55:13
%S A325361 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 A325361 31,32,33,34,35,37,38,39,41,43,46,47,49,50,51,53,54,55,57,58,59,61,62,
%U A325361 64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87,89
%N A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.
%C A325361 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325361 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 A325361 The enumeration of these partitions by sum is given by A320466.
%H A325361 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325361 Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
%e A325361    12: {1,1,2}
%e A325361    20: {1,1,3}
%e A325361    24: {1,1,1,2}
%e A325361    28: {1,1,4}
%e A325361    36: {1,1,2,2}
%e A325361    40: {1,1,1,3}
%e A325361    42: {1,2,4}
%e A325361    44: {1,1,5}
%e A325361    45: {2,2,3}
%e A325361    48: {1,1,1,1,2}
%e A325361    52: {1,1,6}
%e A325361    56: {1,1,1,4}
%e A325361    60: {1,1,2,3}
%e A325361    63: {2,2,4}
%e A325361    66: {1,2,5}
%e A325361    68: {1,1,7}
%e A325361    72: {1,1,1,2,2}
%e A325361    76: {1,1,8}
%e A325361    78: {1,2,6}
%e A325361    80: {1,1,1,1,3}
%t A325361 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325361 Select[Range[100],GreaterEqual@@Differences[primeptn[#]]&]
%Y A325361 Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.
%K A325361 nonn
%O A325361 1,2
%A A325361 _Gus Wiseman_, May 02 2019