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A333146
Number of non-unimodal negated permutations of the multiset of prime indices of n.
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 8, 0, 0, 0
OFFSET
1,24
COMMENTS
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.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
LINKS
Eric Weisstein's World of Mathematics, Unimodal Sequence
FORMULA
a(n) + A333145(n) = A008480(n).
EXAMPLE
The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 96:
(121) (1121) (1212) (1132) (11212) (1232) (111121)
(1211) (1221) (1213) (11221) (1322) (111211)
(2121) (1231) (12112) (2132) (112111)
(1312) (12121) (2231) (121111)
(1321) (12211) (2312)
(2131) (21121) (2321)
(2311) (21211)
(3121)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]], !unimodQ[-#]&]], {n, 30}]
CROSSREFS
Dominated by A008480.
The non-negated version is A332671.
A more interesting version is A332742.
The complement is counted by A333145.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Compositions whose negation is unimodal are A332578.
Partitions with unimodal negated run-lengths are A332638.
Numbers with non-unimodal negated unsorted prime signature are A332642.
Sequence in context: A003475 A248639 A293959 * A135767 A208575 A355037
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 09 2020
STATUS
approved