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A332287
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Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.
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17
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36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561
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OFFSET
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1,1
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COMMENTS
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A sequence of 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), which 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:
36: {1,1,2,2}
50: {1,3,3}
70: {1,3,4}
72: {1,1,1,2,2}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
154: {1,4,5}
180: {1,1,2,2,3}
182: {1,4,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
216: {1,1,1,2,2,2}
225: {2,2,3,3}
242: {1,5,5}
250: {1,3,3,3}
252: {1,1,2,2,4}
280: {1,1,1,3,4}
For example, the prime indices of 70 with 0 appended are (4,3,1,0), with differences (-1,-2,-1), which is not unimodal, so 70 belongs to 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[Append[Reverse[primeMS[#]], 0]]]&]
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CROSSREFS
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The enumeration of these partitions by sum is A332284.
Not assuming the last part is zero gives A332725.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Cf. A001523, A007052, A332280, A332282, 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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