A333145 - OEIS

A333145

Number of unimodal negated permutations of the multiset of prime indices of n.

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, 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, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2

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OFFSET

1,6

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.

Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).

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

LINKS

Table of n, a(n) for n=1..87.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Wikipedia, Permutation pattern

FORMULA

a(n) + A333146(n) = A008480(n).

a(n) = A000005(A028234(n)). - Gus Wiseman, Apr 14 2022

a(n) = A000005(n) - A069157(n). - Gus Wiseman, Apr 14 2022

EXAMPLE

The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:

(1) (12) (122) (123) (1223) (12222) (1234) (12233)

(21) (212) (213) (2123) (21222) (2134) (21233)

(221) (312) (2213) (22122) (3124) (22133)

(321) (3122) (22212) (3214) (31223)

(3212) (22221) (4123) (32123)

(3221) (4213) (32213)

(4312) (33122)

(4321) (33212)

(33221)

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 complementary divisors are counted by A069157.

The non-negated version is A332288.

A more interesting version is A332741.

The complement is counted by A333146.

A001523 counts unimodal compositions.

A007052 counts unimodal normal sequences.

A028233 gives the highest power of the least prime factor, quotient A028234.

A332578 counts compositions whose negation is unimodal.

A332638 counts partitions with unimodal negated run-lengths.

A332642 lists numbers with non-unimodal negated unsorted prime signature.

Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.

Sequence in context: A305150 A351202 A374959 * A335449 A318464 A051265

Adjacent sequences: A333142 A333143 A333144 * A333146 A333147 A333148

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 09 2020

STATUS

approved