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

%I A332643 #9 Feb 16 2025 08:33:59
%S A332643 2100,3300,3900,4200,4410,5100,5700,6468,6600,6900,7644,7800,8400,
%T A332643 8700,9300,9996,10200,10500,10780,10890,11100,11172,11400,12300,12740,
%U A332643 12900,12936,13200,13230,13524,13800,14100,15210,15246,15288,15600,15900,16500,16660
%N A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.
%C A332643 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332643 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%H A332643 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332643 Intersection of A332282 and A332642.
%e A332643 The sequence of terms together with their prime indices begins:
%e A332643    2100: {1,1,2,3,3,4}
%e A332643    3300: {1,1,2,3,3,5}
%e A332643    3900: {1,1,2,3,3,6}
%e A332643    4200: {1,1,1,2,3,3,4}
%e A332643    4410: {1,2,2,3,4,4}
%e A332643    5100: {1,1,2,3,3,7}
%e A332643    5700: {1,1,2,3,3,8}
%e A332643    6468: {1,1,2,4,4,5}
%e A332643    6600: {1,1,1,2,3,3,5}
%e A332643    6900: {1,1,2,3,3,9}
%e A332643    7644: {1,1,2,4,4,6}
%e A332643    7800: {1,1,1,2,3,3,6}
%e A332643    8400: {1,1,1,1,2,3,3,4}
%e A332643    8700: {1,1,2,3,3,10}
%e A332643    9300: {1,1,2,3,3,11}
%e A332643    9996: {1,1,2,4,4,7}
%e A332643   10200: {1,1,1,2,3,3,7}
%e A332643   10500: {1,1,2,3,3,3,4}
%e A332643   10780: {1,1,3,4,4,5}
%e A332643   10890: {1,2,2,3,5,5}
%t A332643 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332643 Select[Range[10000],!unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]
%Y A332643 Not requiring non-unimodal negation gives A332282.
%Y A332643 These are the Heinz numbers of the partitions counted by A332640.
%Y A332643 Not requiring non-unimodality gives A332642.
%Y A332643 The case of compositions is A332870.
%Y A332643 Unimodal compositions are A001523.
%Y A332643 Non-unimodal permutations are A059204.
%Y A332643 Non-unimodal compositions are A115981.
%Y A332643 Unsorted prime signature is A124010.
%Y A332643 Non-unimodal normal sequences are A328509.
%Y A332643 Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.
%Y A332643 Compositions whose negation is unimodal are A332578.
%Y A332643 Compositions whose negation is not unimodal are A332669.
%Y A332643 Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
%Y A332643 Numbers whose signature is neither increasing nor decreasing are A332831.
%Y A332643 Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.
%K A332643 nonn
%O A332643 1,1
%A A332643 _Gus Wiseman_, Feb 28 2020