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A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.

%I A325397 #7 May 03 2019 08:37:21
%S A325397 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 A325397 31,32,33,34,35,37,38,39,41,43,46,47,49,50,51,53,54,55,57,58,59,61,62,
%U A325397 64,65,67,69,70,71,73,74,75,77,79,81,82,83,85,86,87,89
%N A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0.
%C A325397 First differs from A325361 in lacking 150.
%C A325397 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325397 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 A325397 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 A325397 The enumeration of these partitions by sum is given by A325353.
%H A325397 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325397 Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
%e A325397    12: {1,1,2}
%e A325397    20: {1,1,3}
%e A325397    24: {1,1,1,2}
%e A325397    28: {1,1,4}
%e A325397    36: {1,1,2,2}
%e A325397    40: {1,1,1,3}
%e A325397    42: {1,2,4}
%e A325397    44: {1,1,5}
%e A325397    45: {2,2,3}
%e A325397    48: {1,1,1,1,2}
%e A325397    52: {1,1,6}
%e A325397    56: {1,1,1,4}
%e A325397    60: {1,1,2,3}
%e A325397    63: {2,2,4}
%e A325397    66: {1,2,5}
%e A325397    68: {1,1,7}
%e A325397    72: {1,1,1,2,2}
%e A325397    76: {1,1,8}
%e A325397    78: {1,2,6}
%e A325397    80: {1,1,1,1,3}
%e A325397 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 A325397 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325397 Select[Range[100],And@@Table[GreaterEqual@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
%Y A325397 Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467.
%K A325397 nonn
%O A325397 1,2
%A A325397 _Gus Wiseman_, May 02 2019