A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 6, 1, 5, 3, 3, 1, 9, 2, 3, 4, 5, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 9, 1, 7, 1, 5, 5, 3, 1, 17, 2, 6, 3, 5, 1, 12, 3, 9, 3, 3, 1, 11, 1, 3, 5, 32, 3, 7, 1, 5, 3, 7, 1, 18, 1, 3, 6, 5, 3, 7, 1, 17, 8, 3, 1, 11

Offset: 1

Author

Gus Wiseman, Apr 17 2020

Keywords

• nonn

Comments

Unsorted prime signature (A124010) 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 unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.
		

Crossrefs

Positions of first appearances are A057335 (a partial inverse).

Least number with same prime signature is A071364.

Unsorted prime signature is A124010.

Least number with reversed prime signature is A331580.

Minimal numbers with standard reversed prime signatures are A334031.

The reversed version is A334033.

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.

- Permutations are A333218.

- Heinz number is A333219.

Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.

Programs

Mathematica

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]

Formula

a(A057335(n)) = n.

A057335(a(n)) = A071364(n).

a(A334031(n))= A059893(n).

A334031(a(n)) = A331580(n).