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

Offset: 1

Author

Gus Wiseman, Mar 09 2020

Keywords

• nonn

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

Examples

			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)
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence
• Wikipedia, Permutation pattern

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.

Programs

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}]

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