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A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal. 11

%I #8 Feb 28 2020 22:55:00

%S 2100,3300,3900,4200,4410,5100,5700,6468,6600,6900,7644,7800,8400,

%T 8700,9300,9996,10200,10500,10780,10890,11100,11172,11400,12300,12740,

%U 12900,12936,13200,13230,13524,13800,14100,15210,15246,15288,15600,15900,16500,16660

%N Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%F Intersection of A332282 and A332642.

%e The sequence of terms together with their prime indices begins:

%e 2100: {1,1,2,3,3,4}

%e 3300: {1,1,2,3,3,5}

%e 3900: {1,1,2,3,3,6}

%e 4200: {1,1,1,2,3,3,4}

%e 4410: {1,2,2,3,4,4}

%e 5100: {1,1,2,3,3,7}

%e 5700: {1,1,2,3,3,8}

%e 6468: {1,1,2,4,4,5}

%e 6600: {1,1,1,2,3,3,5}

%e 6900: {1,1,2,3,3,9}

%e 7644: {1,1,2,4,4,6}

%e 7800: {1,1,1,2,3,3,6}

%e 8400: {1,1,1,1,2,3,3,4}

%e 8700: {1,1,2,3,3,10}

%e 9300: {1,1,2,3,3,11}

%e 9996: {1,1,2,4,4,7}

%e 10200: {1,1,1,2,3,3,7}

%e 10500: {1,1,2,3,3,3,4}

%e 10780: {1,1,3,4,4,5}

%e 10890: {1,2,2,3,5,5}

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]

%t Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

%Y Not requiring non-unimodal negation gives A332282.

%Y These are the Heinz numbers of the partitions counted by A332640.

%Y Not requiring non-unimodality gives A332642.

%Y The case of compositions is A332870.

%Y Unimodal compositions are A001523.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Unsorted prime signature is A124010.

%Y Non-unimodal normal sequences are A328509.

%Y Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.

%Y Compositions whose negation is unimodal are A332578.

%Y Compositions whose negation is not unimodal are A332669.

%Y Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.

%Y Numbers whose signature is neither increasing nor decreasing are A332831.

%Y Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 28 2020

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)