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A332642 Numbers whose negated unsorted prime signature is not unimodal.

%I A332642 #11 Feb 16 2025 08:33:59
%S A332642 90,126,198,234,270,306,342,350,378,414,522,525,540,550,558,594,630,
%T A332642 650,666,702,738,756,774,810,825,846,850,918,950,954,975,990,1026,
%U A332642 1050,1062,1078,1098,1134,1150,1170,1188,1206,1242,1274,1275,1278,1314,1350,1386
%N A332642 Numbers whose negated unsorted prime signature is not unimodal.
%C A332642 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332642 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%H A332642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332642 The sequence of terms together with their prime indices begins:
%e A332642     90: {1,2,2,3}
%e A332642    126: {1,2,2,4}
%e A332642    198: {1,2,2,5}
%e A332642    234: {1,2,2,6}
%e A332642    270: {1,2,2,2,3}
%e A332642    306: {1,2,2,7}
%e A332642    342: {1,2,2,8}
%e A332642    350: {1,3,3,4}
%e A332642    378: {1,2,2,2,4}
%e A332642    414: {1,2,2,9}
%e A332642    522: {1,2,2,10}
%e A332642    525: {2,3,3,4}
%e A332642    540: {1,1,2,2,2,3}
%e A332642    550: {1,3,3,5}
%e A332642    558: {1,2,2,11}
%e A332642    594: {1,2,2,2,5}
%e A332642    630: {1,2,2,3,4}
%e A332642    650: {1,3,3,6}
%e A332642    666: {1,2,2,12}
%e A332642    702: {1,2,2,2,6}
%e A332642 For example, 630 has negated unsorted prime signature (-1,-2,-1,-1), which is not unimodal, so 630 is in the sequence.
%t A332642 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332642 Select[Range[2000],!unimodQ[-Last/@FactorInteger[#]]&]
%Y A332642 These are the Heinz numbers of the partitions counted by A332639.
%Y A332642 The case that is not unimodal either is A332643.
%Y A332642 The version for compositions is A332669.
%Y A332642 The complement is A332282.
%Y A332642 Unimodal compositions are A001523.
%Y A332642 Non-unimodal permutations are A059204.
%Y A332642 Non-unimodal compositions are A115981.
%Y A332642 Unsorted prime signature is A124010.
%Y A332642 Non-unimodal normal sequences are A328509.
%Y A332642 The number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n is A332742(n).
%Y A332642 Partitions whose negated 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.
%Y A332642 Cf. A007052, A056239, A112798, A181821, A242031, A329747, A332280, A332281, A332578, A332671, A332831.
%K A332642 nonn
%O A332642 1,1
%A A332642 _Gus Wiseman_, Feb 28 2020