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 A253565 #32 Dec 25 2022 19:11:13
%S A253565 1,2,3,4,5,9,6,8,7,25,15,27,10,18,12,16,11,49,35,125,21,75,45,81,14,
%T A253565 50,30,54,20,36,24,32,13,121,77,343,55,245,175,625,33,147,105,375,63,
%U A253565 225,135,243,22,98,70,250,42,150,90,162,28,100,60,108,40,72,48,64,17,169,143,1331,91,847,539,2401,65,605,385,1715,275,1225,875,3125,39
%N A253565 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).
%C A253565 This sequence can be represented as a binary tree. Each child to the left is obtained by applying A253550 to the parent, and each child to the right is obtained by applying A253560 to the parent:
%C A253565 1
%C A253565 |
%C A253565 ...................2...................
%C A253565 3 4
%C A253565 5......../ \........9 6......../ \........8
%C A253565 / \ / \ / \ / \
%C A253565 / \ / \ / \ / \
%C A253565 / \ / \ / \ / \
%C A253565 7 25 15 27 10 18 12 16
%C A253565 11 49 35 125 21 75 45 81 14 50 30 54 20 36 24 32
%C A253565 etc.
%C A253565 Sequence A253563 is the mirror image of the same tree. Also in binary trees A005940 and A163511 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees. Of these four trees, this is the one where the left child is always smaller than the right child.
%C A253565 Note that the indexing of sequence starts from 0, although its range starts from one.
%C A253565 The term a(n) is the Heinz number of the adjusted partial sums of the n-th composition in standard order, where (1) the k-th composition in standard order (graded reverse-lexicographic, 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, (2) the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and (3) we define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts. See formula for a simplification. A triangular form is A242628. The inverse is A253566. The non-adjusted version is A358170. - _Gus Wiseman_, Dec 17 2022
%H A253565 Antti Karttunen, <a href="/A253565/b253565.txt">Table of n, a(n) for n = 0..8191</a>
%H A253565 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A253565 a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).
%F A253565 As a composition of related permutations:
%F A253565 a(n) = A122111(A163511(n)).
%F A253565 a(n) = A253563(A054429(n)).
%F A253565 Other identities and observations. For all n >= 0:
%F A253565 a(2n+1) - a(2n) > 0. [See the comment above.]
%F A253565 If n = 2^(x_1)+...+2^(x_k) then a(n) = Product_{i=1..k} prime(x_k-x_{i-1}-k+i) where x_0 = 0. - _Gus Wiseman_, Dec 23 2022
%e A253565 From _Gus Wiseman_, Dec 23 2022: (Start)
%e A253565 This represents the following bijection between compositions and partitions. The n-th composition in standard order together with the reversed prime indices of a(n) are:
%e A253565 0: () -> ()
%e A253565 1: (1) -> (1)
%e A253565 2: (2) -> (2)
%e A253565 3: (1,1) -> (1,1)
%e A253565 4: (3) -> (3)
%e A253565 5: (2,1) -> (2,2)
%e A253565 6: (1,2) -> (2,1)
%e A253565 7: (1,1,1) -> (1,1,1)
%e A253565 8: (4) -> (4)
%e A253565 9: (3,1) -> (3,3)
%e A253565 10: (2,2) -> (3,2)
%e A253565 11: (2,1,1) -> (2,2,2)
%e A253565 12: (1,3) -> (3,1)
%e A253565 13: (1,2,1) -> (2,2,1)
%e A253565 14: (1,1,2) -> (2,1,1)
%e A253565 15: (1,1,1,1) -> (1,1,1,1)
%e A253565 (End)
%t A253565 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A253565 Times@@Prime/@#&/@Table[Accumulate[stc[n]-1]+1,{n,0,60}] (* _Gus Wiseman_, Dec 17 2022 *)
%o A253565 (Scheme, two versions)
%o A253565 (definec (A253565 n) (cond ((< n 2) (+ 1 n)) ((even? n) (A253550 (A253565 (/ n 2)))) (else (A253560 (A253565 (/ (- n 1) 2))))))
%o A253565 (define (A253565 n) (A122111 (A163511 n)))
%Y A253565 Inverse: A253566.
%Y A253565 Cf. A252737 (row sums), A252738 (row products).
%Y A253565 Cf. A122111, A163511, A253550, A253560, A252464, A253563, A054429.
%Y A253565 Applying A001222 gives A000120.
%Y A253565 A reverse version is A005940.
%Y A253565 These are the Heinz numbers of the rows of A242628.
%Y A253565 Sum of prime indices of a(n) is A359043, reverse A161511.
%Y A253565 A048793 gives partial sums of reversed standard comps, Heinz number A019565.
%Y A253565 A066099 lists standard compositions.
%Y A253565 A112798 list prime indices, sum A056239.
%Y A253565 A358134 gives partial sums of standard compositions, Heinz number A358170.
%Y A253565 Cf. A029837, A059893, A070939, A241916, A358135, A358137, A358195, A359042.
%K A253565 nonn,tabf,look
%O A253565 0,2
%A A253565 _Antti Karttunen_, Jan 03 2015