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A332642
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Numbers whose negated unsorted prime signature is not unimodal.
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33
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90, 126, 198, 234, 270, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 594, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1188, 1206, 1242, 1274, 1275, 1278, 1314, 1350, 1386
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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.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
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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}
198: {1,2,2,5}
234: {1,2,2,6}
270: {1,2,2,2,3}
306: {1,2,2,7}
342: {1,2,2,8}
350: {1,3,3,4}
378: {1,2,2,2,4}
414: {1,2,2,9}
522: {1,2,2,10}
525: {2,3,3,4}
540: {1,1,2,2,2,3}
550: {1,3,3,5}
558: {1,2,2,11}
594: {1,2,2,2,5}
630: {1,2,2,3,4}
650: {1,3,3,6}
666: {1,2,2,12}
702: {1,2,2,2,6}
For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.
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MATHEMATICA
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unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Select[Range[2000], !unimodQ[-Last/@FactorInteger[#]]&]
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CROSSREFS
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These are the Heinz numbers of the partitions counted by A332639.
The case that is not unimodal either is A332643.
The version for compositions is A332669.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unsorted prime signature is A124010.
Non-unimodal normal sequences are A328509.
The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).
Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.
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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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