A334033 The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 6, 1, 3, 3, 8, 1, 5, 1, 6, 3, 3, 1, 12, 2, 3, 4, 6, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 12, 1, 7, 1, 6, 6, 3, 1, 24, 2, 5, 3, 6, 1, 9, 3, 12, 3, 3, 1, 14, 1, 3, 6, 32, 3, 7, 1, 6, 3, 7, 1, 20, 1, 3, 5, 6, 3, 7, 1, 24, 8, 3, 1

Offset

1,4

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.

Links

Table of n, a(n) for n=1..83.

Formula

a(A334031(n)) = n.

A334031(a(n)) = A071364(n).

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

A057335(a(n)) = A331580(n).

Example

The unsorted prime signature of 12345678 is (1,2,1,1), whose reverse (1,1,2,1) is the 29th composition in standard order, so a(12345678) = 29.

Mathematica

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

Crossrefs

Positions of first appearances are A334031.

The non-reversed version is A334032.

Unsorted prime signature is A124010.

Least number with reversed prime signature is A331580.

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, A057335, A071364, A112798, A124767, A228351, A233249, A329139, A333220.

Sequence in context: A334032 A366191 A097283 * A339564 A296119 A300836

Adjacent sequences: A334030 A334031 A334032 * A334034 A334035 A334036

Keyword

nonn

Author

Gus Wiseman, Apr 18 2020

Status

approved