

A334032


The a(n)th composition in standard order (graded reverselexicographic) is the unsorted prime signature of n.


5



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
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OFFSET

1,4


COMMENTS

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.
The kth 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..84.


FORMULA

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(n).


EXAMPLE

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


MATHEMATICA

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


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.
Sequence in context: A079616 A292587 A336571 * A097283 A334033 A296119
Adjacent sequences: A334029 A334030 A334031 * A334033 A334034 A334035


KEYWORD

nonn


AUTHOR

Gus Wiseman, Apr 17 2020


STATUS

approved



