A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 121, 124, 125, 126, 128, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172

Offset: 1

Author

Gus Wiseman, Dec 23 2021

Keywords

• nonn

Comments

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

Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.

Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.

Examples

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
		

Crossrefs

This is the nonsquarefree case of the complement of A349794.

These are the Heinz numbers of the partitions counted by A349795.

A version for compositions is A349799, counted by A349800.

A complementary version is A350140, counted by A349796.

A001250 = alternating permutations, ranked by A349051, complement A348615.

A005117 = squarefree numbers, complement A013929.

A025047/A025048/A025049 = alternating compositions, ranked by A345167.

A056239 adds up prime indices, row sums of A112798, row lengths A001222.

A124010 = prime signature, sorted A118914.

A345164 = alternating permutations of prime indices, complement A350251.

A349052/A129852/A129853 = weakly alternating compositions.

A349053 = non-weakly alternating compositions, ranked by A349057.

A349056 = weakly alternating permutations of prime indices.

A349058 = weakly alternating patterns, complement A350138.

A349060 = weakly alternating partitions, complement A349061.

Cf. A000111, A096441, A117298, A335433, A335448, A335452, A344614, A344652, A344653, A345173, A349059, A349797.

Programs

Mathematica

Select[Range[100],!SquareFreeQ[#]&&(PrimePowerQ[#]||And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}])&]