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A334033 The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A336571 A334032 A097283 * A296119 A300836 A118314

Adjacent sequences:  A334030 A334031 A334032 * A334034 A334035 A334036

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 18 2020

STATUS

approved

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Last modified September 18 23:17 EDT 2020. Contains 337175 sequences. (Running on oeis4.)