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A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.

%I A325400 #4 May 03 2019 08:37:46
%S A325400 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 A325400 28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,
%U A325400 53,55,56,57,58,59,61,62,63,64,65,66,67,68,69,71,73,74
%N A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.
%C A325400 First differs from A109427 in lacking 54.
%C A325400 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325400 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 A325400 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 A325400 The enumeration of these partitions by sum is given by A325354.
%H A325400 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325400 Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
%e A325400    18: {1,2,2}
%e A325400    36: {1,1,2,2}
%e A325400    50: {1,3,3}
%e A325400    54: {1,2,2,2}
%e A325400    60: {1,1,2,3}
%e A325400    70: {1,3,4}
%e A325400    72: {1,1,1,2,2}
%e A325400    75: {2,3,3}
%e A325400    90: {1,2,2,3}
%e A325400    98: {1,4,4}
%e A325400   100: {1,1,3,3}
%e A325400   108: {1,1,2,2,2}
%e A325400   120: {1,1,1,2,3}
%e A325400   126: {1,2,2,4}
%e A325400   140: {1,1,3,4}
%e A325400   144: {1,1,1,1,2,2}
%e A325400   147: {2,4,4}
%e A325400   150: {1,2,3,3}
%e A325400   154: {1,4,5}
%e A325400   162: {1,2,2,2,2}
%t A325400 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325400 Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]
%Y A325400 Cf. A007294, A056239, A112798, A240026, A325354, A325360, A325362, A325394, A325397, A325398, A325399, A325405, A325467.
%K A325400 nonn
%O A325400 1,2
%A A325400 _Gus Wiseman_, May 02 2019