A332643 Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.

2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660

Offset

1,1

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Links

Table of n, a(n) for n=1..39.

MathWorld, Unimodal Sequence

Formula

Intersection of A332282 and A332642.

Example

The sequence of terms together with their prime indices begins:
   2100: {1,1,2,3,3,4}
   3300: {1,1,2,3,3,5}
   3900: {1,1,2,3,3,6}
   4200: {1,1,1,2,3,3,4}
   4410: {1,2,2,3,4,4}
   5100: {1,1,2,3,3,7}
   5700: {1,1,2,3,3,8}
   6468: {1,1,2,4,4,5}
   6600: {1,1,1,2,3,3,5}
   6900: {1,1,2,3,3,9}
   7644: {1,1,2,4,4,6}
   7800: {1,1,1,2,3,3,6}
   8400: {1,1,1,1,2,3,3,4}
   8700: {1,1,2,3,3,10}
   9300: {1,1,2,3,3,11}
   9996: {1,1,2,4,4,7}
  10200: {1,1,1,2,3,3,7}
  10500: {1,1,2,3,3,3,4}
  10780: {1,1,3,4,4,5}
  10890: {1,2,2,3,5,5}

Mathematica

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Select[Range[10000], !unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]

Crossrefs

Not requiring non-unimodal negation gives A332282.

These are the Heinz numbers of the partitions counted by A332640.

Not requiring non-unimodality gives A332642.

The case of compositions is A332870.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Unsorted prime signature is A124010.

Non-unimodal normal sequences are A328509.

Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287.

Compositions whose negation is unimodal are A332578.

Compositions whose negation is not unimodal are A332669.

Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832.

Numbers whose signature is neither increasing nor decreasing are A332831.

Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.

Sequence in context: A045051 A102504 A270761 * A015293 A159812 A233729

Adjacent sequences: A332640 A332641 A332642 * A332644 A332645 A332646

Keyword

nonn

Author

Gus Wiseman, Feb 28 2020

Status

approved