A349794 Numbers whose prime signature has an odd term other than the first or last.

30, 42, 60, 66, 70, 78, 84, 102, 105, 110, 114, 120, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 182, 186, 190, 195, 204, 210, 220, 222, 228, 230, 231, 238, 240, 246, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285, 286, 290, 294, 300, 308

Offset: 1

Author

Gus Wiseman, Dec 06 2021

Keywords

• nonn

Comments

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

Also numbers whose multiset of prime factors is not weakly alternating, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict decreases are allowed.

Examples

			The terms and their prime indices begin:
   30: {1,2,3}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}
		

Crossrefs

The complement for compositions is A025047, ranked by A345167.

Signatures of this type are counted by A274230, complement A027383.

The strong case is A289553, complement A167171.

The strong case for compositions is A345192, ranked by A345168.

The version for compositions is A349053, ranked by A349057.

These partitions are counted by A349061, complement A349060, strong A349801.

The non-strict case is counted by A349795.

A001250 counts alternating permutations, complement A348615.

A096441 counts weakly alternating partitions if 0 is appended.

A345164 counts alternating permutations of prime indices, weak A349056.

A345170 counts partitions w/ an alternating permutation, ranked by A345172.

A349052 counts weakly alternating compositions.

A349059 counts weakly alternating ordered factorizations, strong A348610.

Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.

Programs

Mathematica

Select[Range[100],PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]