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%I #5 Mar 02 2020 18:54:51
%S 12,20,24,28,36,40,44,45,48,52,56,60,63,68,72,76,80,84,88,90,92,96,99,
%T 100,104,108,112,116,117,120,124,126,132,135,136,140,144,148,152,153,
%U 156,160,164,165,168,171,172,175,176,180,184,188,189,192,195,196,198
%N Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.
%C First differs from A065201 in having 165.
%C First differs from A316597 in having 36.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e The sequence of 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 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 68: {1,1,7}
%e 72: {1,1,1,2,2}
%e 76: {1,1,8}
%e 80: {1,1,1,1,3}
%e 84: {1,1,2,4}
%e 88: {1,1,1,5}
%e 90: {1,2,2,3}
%e For example, 60 is the Heinz number of (3,2,1,1), with negated 0-appended first-differences (1,1,0,1), which are not unimodal, so 60 is in the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Select[Range[100],!unimodQ[Differences[Prepend[primeMS[#],0]]]&]
%Y The non-negated version is A332287.
%Y The version for of run-lengths (instead of differences) is A332642.
%Y The enumeration of these partitions by sum is A332744.
%Y Unimodal compositions are A001523.
%Y Non-unimodal compositions are A115981.
%Y Heinz numbers of partitions with non-unimodal run-lengths are A332282.
%Y Partitions whose 0-appended first differences are unimodal are A332283.
%Y Compositions whose negation is unimodal are A332578.
%Y Compositions whose negation is not unimodal are A332669.
%Y Cf. A059204, A227038, A332284, A332285, A332286, A332578, A332638, A332639, A332670, A332725, A332728.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 02 2020
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