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

0,2

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.

Links

Table of n, a(n) for n=0..53.

Formula

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

Example

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.

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}]

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.

Sequence in context: A122408 A326037 A100055 * A304660 A344902 A104001

Adjacent sequences: A334028 A334029 A334030 * A334032 A334033 A334034

Keyword

nonn

Author

Gus Wiseman, Apr 17 2020

Status

approved