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%I #10 May 28 2020 05:01:39
%S 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,
%T 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,
%U 1,5,3,7,1,18,1,3,6,5,3,7,1,17,8,3,1,11
%N The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.
%C Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.
%C 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.
%F a(A057335(n)) = n.
%F A057335(a(n)) = A071364(n).
%F a(A334031(n))= A059893(n).
%F A334031(a(n)) = A331580(n).
%e The unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.
%t stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
%t Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]
%Y Positions of first appearances are A057335 (a partial inverse).
%Y Least number with same prime signature is A071364.
%Y Unsorted prime signature is A124010.
%Y Least number with reversed prime signature is A331580.
%Y Minimal numbers with standard reversed prime signatures are A334031.
%Y The reversed version is A334033.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - Length is A000120.
%Y - Sum is A070939.
%Y - Strict compositions are A233564.
%Y - Constant compositions are A272919.
%Y - Aperiodic compositions are A328594.
%Y - Normal compositions are A333217.
%Y - Permutations are A333218.
%Y - Heinz number is A333219.
%Y Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.
%K nonn
%O 1,4
%A _Gus Wiseman_, Apr 17 2020
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