This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253566 #17 Dec 28 2022 09:04:50
%S A253566 0,1,2,3,4,6,8,7,5,12,16,14,32,24,10,15,64,13,128,28,20,48,256,30,9,
%T A253566 96,11,56,512,26,1024,31,40,192,18,29,2048,384,80,60,4096,52,8192,112,
%U A253566 22,768,16384,62,17,25,160,224,32768,27,36,120,320,1536,65536,58,131072,3072,44,63,72,104,262144,448,640,50,524288,61,1048576,6144,21
%N A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).
%C A253566 Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564.
%C A253566 The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - _Gus Wiseman_, Dec 23 2022
%H A253566 Antti Karttunen, <a href="/A253566/b253566.txt">Table of n, a(n) for n = 1..1024</a>
%H A253566 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A253566 a(n) = A243071(A122111(n)).
%F A253566 As a composition of other permutations:
%F A253566 a(n) = A054429(A253564(n)).
%F A253566 a(n) = A336120(n) + A336125(n). - _Antti Karttunen_, Jul 18 2020
%F A253566 If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - _Gus Wiseman_, Dec 23 2022
%e A253566 From _Gus Wiseman_, Dec 23 2022: (Start)
%e A253566 This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
%e A253566 1: () -> ()
%e A253566 2: (1) -> (1)
%e A253566 3: (2) -> (2)
%e A253566 4: (1,1) -> (1,1)
%e A253566 5: (3) -> (3)
%e A253566 6: (2,1) -> (1,2)
%e A253566 7: (4) -> (4)
%e A253566 8: (1,1,1) -> (1,1,1)
%e A253566 9: (2,2) -> (2,1)
%e A253566 10: (3,1) -> (1,3)
%e A253566 11: (5) -> (5)
%e A253566 12: (2,1,1) -> (1,1,2)
%e A253566 13: (6) -> (6)
%e A253566 14: (4,1) -> (1,4)
%e A253566 15: (3,2) -> (2,2)
%e A253566 16: (1,1,1,1) -> (1,1,1,1)
%e A253566 (End)
%t A253566 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A253566 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A253566 stcinv/@Table[Differences[Prepend[primeMS[n],1]]+1,{n,100}] (* _Gus Wiseman_, Dec 23 2022 *)
%o A253566 (Scheme) (define (A253566 n) (A243071 (A122111 n)))
%Y A253566 Inverse: A253565.
%Y A253566 Cf. A122111, A243071, A253564, A054429, A336120, A336125.
%Y A253566 Applying A000120 gives A001222.
%Y A253566 A reverse version is A156552, inverse essentially A005940.
%Y A253566 The inverse is A253565, triangular form A242628.
%Y A253566 The triangular form is A358169.
%Y A253566 A048793 gives partial sums of reversed standard comps, Heinz number A019565.
%Y A253566 A066099 lists standard compositions, lengths A000120, sums A070939.
%Y A253566 A112798 list prime indices, sum A056239.
%Y A253566 A358134 gives partial sums of standard compositions, Heinz number A358170.
%Y A253566 Cf. A029837, A241916, A243055, A287352, A325390, A355534, A358171, A359042.
%K A253566 nonn
%O A253566 1,3
%A A253566 _Antti Karttunen_, Jan 03 2015