A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

1, 2, 4, 6, 8, 18, 12, 30, 16, 54, 36, 150, 24, 90, 60, 210, 32, 162, 108, 750, 72, 450, 300, 1470, 48, 270, 180, 1050, 120, 630, 420, 2310, 64, 486, 324, 3750, 216, 2250, 1500, 10290, 144, 1350, 900, 7350, 600, 4410, 2940, 25410, 96, 810, 540, 5250, 360, 3150

Offset: 0

Author

Gus Wiseman, Apr 17 2020

Keywords

• nonn

Comments

All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.

Unsorted prime signature is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       4: {1,1}
       6: {1,2}
       8: {1,1,1}
      18: {1,2,2}
      12: {1,1,2}
      30: {1,2,3}
      16: {1,1,1,1}
      54: {1,2,2,2}
      36: {1,1,2,2}
     150: {1,2,3,3}
      24: {1,1,1,2}
      90: {1,2,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
      32: {1,1,1,1,1}
     162: {1,2,2,2,2}
For example, the 13th composition in standard order is (1,2,1), and the least number with prime signature (1,2,1) is 90 = 2^1 * 3^2 * 5^1, so a(13) = 90.
		

Crossrefs

The range is A055932.

The non-reversed version is A057335.

Unsorted prime signature is A124010.

Numbers whose prime signature is aperiodic are A329139.

Normal numbers with standard compositions as prime signature are A334032.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Sum is A070939.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Aperiodic compositions are A328594.

- Normal compositions are A333217.

- Heinz number is A333219.

Cf. A029931, A048793, A056239, A112798, A228351, A233249, A272020, A333218, A333220, A334032.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Product[Prime[i]^stc[n][[-i]],{i,DigitCount[n,2,1]}],{n,0,100}]

Formula

a(n) = A057335(A059893(n)).