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%I #14 May 15 2022 11:42:23
%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,3,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,2,
%T 2,3,1,2,2,2,1,4,1,2,2,2,1,2,1,3,2,2,1,4,2,2,2,2,1,4,1,2,2,1,2,4,1,2,
%U 2,4,1,3,1,2,3,2,2,4,1,2,1,2,1,4,2,2,2
%N Number of unimodal negated permutations of the multiset of prime indices of n.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
%C Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - _Gus Wiseman_, Apr 12 2022
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%F a(n) + A333146(n) = A008480(n).
%F a(n) = A000005(A028234(n)). - _Gus Wiseman_, Apr 14 2022
%F a(n) = A000005(n) - A069157(n). - _Gus Wiseman_, Apr 14 2022
%e The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
%e (1) (12) (122) (123) (1223) (12222) (1234) (12233)
%e (21) (212) (213) (2123) (21222) (2134) (21233)
%e (221) (312) (2213) (22122) (3124) (22133)
%e (321) (3122) (22212) (3214) (31223)
%e (3212) (22221) (4123) (32123)
%e (3221) (4213) (32213)
%e (4312) (33122)
%e (4321) (33212)
%e (33221)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Table[Length[Select[Permutations[primeMS[n]],unimodQ[-#]&]],{n,30}]
%Y Dominated by A008480.
%Y The complementary divisors are counted by A069157.
%Y The non-negated version is A332288.
%Y A more interesting version is A332741.
%Y The complement is counted by A333146.
%Y A001523 counts unimodal compositions.
%Y A007052 counts unimodal normal sequences.
%Y A028233 gives the highest power of the least prime factor, quotient A028234.
%Y A332578 counts compositions whose negation is unimodal.
%Y A332638 counts partitions with unimodal negated run-lengths.
%Y A332642 lists numbers with non-unimodal negated unsorted prime signature.
%Y Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.
%K nonn
%O 1,6
%A _Gus Wiseman_, Mar 09 2020
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