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A332725
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Heinz numbers of integer partitions whose negated first differences are not unimodal.
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3
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90, 126, 180, 198, 234, 252, 270, 306, 342, 350, 360, 378, 396, 414, 450, 468, 504, 522, 525, 540, 550, 558, 594, 612, 630, 650, 666, 684, 700, 702, 720, 738, 756, 774, 792, 810, 825, 828, 846, 850, 882, 900, 910, 918, 936, 950, 954, 975, 990, 1008, 1026, 1044
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OFFSET
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1,1
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COMMENTS
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A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
90: {1,2,2,3}
126: {1,2,2,4}
180: {1,1,2,2,3}
198: {1,2,2,5}
234: {1,2,2,6}
252: {1,1,2,2,4}
270: {1,2,2,2,3}
306: {1,2,2,7}
342: {1,2,2,8}
350: {1,3,3,4}
360: {1,1,1,2,2,3}
378: {1,2,2,2,4}
396: {1,1,2,2,5}
414: {1,2,2,9}
450: {1,2,2,3,3}
468: {1,1,2,2,6}
504: {1,1,1,2,2,4}
522: {1,2,2,10}
525: {2,3,3,4}
540: {1,1,2,2,2,3}
For example, 350 is the Heinz number of (4,3,3,1), with negated first differences (1,0,2), which is not unimodal, so 350 is in the sequence.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Select[Range[1000], !unimodQ[Differences[primeMS[#]]]&]
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CROSSREFS
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The complement is too full.
The enumeration of these partitions by sum is A332284.
The version where the last part is taken to be 0 is A332832.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Heinz numbers of partitions with weakly increasing differences are A325360.
Cf. A001523, A007052, A240026, A332280, A332283, A332285, A332286, A332288, A332294, A332579, A332639, A332642.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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