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 A263267 #48 Mar 18 2020 04:16:01
%S A263267 0,1,2,3,4,6,5,8,9,10,12,7,11,14,18,13,15,16,20,22,17,24,25,26,28,30,
%T A263267 19,21,32,34,23,40,38,42,27,44,48,46,29,36,50,56,60,49,52,54,31,33,72,
%U A263267 58,35,84,62,66,37,39,96,68,70,41,45,104,108,74,76,78,80,43,47,120,81,82,90,88,51,128,132,83,85,86,94,53,55,136,140,87,92,102
%N A263267 Breadth-first traversal of the tree defined by the edge-relation A049820(child) = parent.
%C A263267 It is conjectured that the terms of A259934 trace the only infinite path in this tree.
%C A263267 After the root (0), the tree narrows next time to the width of just one node at level A262508(1) = 9236, with vertex 119143.
%H A263267 Antti Karttunen, <a href="/A263267/b263267.txt">Table of n, a(n) for n = 0..10425; levels 0 .. 1001 of the tree</a>
%H A263267 Michael De Vlieger, <a href="https://oeis.org/A263267/a263267_3.pdf">Poster illustrating A259934 and A263267</a>
%H A263267 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A263267 Rows 0 - 21 of the table. The lines show the nodes of the tree connected by the edge-relation A049820(child) = parent:
%e A263267 0;
%e A263267 | \
%e A263267 1, 2;
%e A263267 | \ \
%e A263267 3, 4, 6;____
%e A263267 | | | \ \
%e A263267 5, 8, 9, 10, 12;
%e A263267 | | | |
%e A263267 7, _ 11, 14, 18;
%e A263267 / | \ \ \
%e A263267 13, 15, 16, 20, 22;____
%e A263267 | | / | \ \
%e A263267 17, 24, 25, 26, 28, 30;
%e A263267 | \ | |
%e A263267 19, 21, 32, 34;
%e A263267 | | | \
%e A263267 23, 40, 38, 42;____
%e A263267 | | \ \
%e A263267 27, 44, 48, 46;____
%e A263267 | \ | | \ | \ \
%e A263267 29, 36, 50, 56, 60, 49, 52, 54;
%e A263267 | \ | |
%e A263267 31, 33, 72, 58;
%e A263267 | | | \
%e A263267 35, 84, 62, 66;
%e A263267 | \ | | \
%e A263267 37, 39, 96, 68, 70;_______
%e A263267 | \ | \ / | \ \
%e A263267 41, 45, 104, 108, 74, 76, 78, 80;
%e A263267 | | | | | \ \
%e A263267 43, 47, 120, _81, 82, 90, 88;
%e A263267 | | \ / | | |
%e A263267 51, 128, 132, 83, 85, 86, 94;
%e A263267 | \ | \ | | |
%e A263267 53, 55 136, 140 87, 92, 102;______
%e A263267 | | \ | | \ \
%e A263267 57,_ 89, 91, 98, 106, 110, 112;
%e A263267 / | \ / / \ | |
%e A263267 59, 63, 64, 93, 95, 100, 114, 116;
%e A263267 | | | | \
%e A263267 61, 99, 97, _118, 126;
%e A263267 | | | / | \
%e A263267 65, 101, 105, 121, 122, 124;
%e A263267 (See also _Michael De Vlieger_'s poster in the Links section.)
%o A263267 (PARI)
%o A263267 uplim = 125753; \\ = A263260(10001).
%o A263267 checklimit = 1440; \\ Hard limit 1440 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
%o A263267 v263267 = vector(uplim);
%o A263267 A263267 = n -> if(!n,n,v263267[n]);
%o A263267 z = 0; for(n=0, uplim, t = A263267(n); write("b263267.txt", n, " ", t); for(k=t+1, t+checklimit, if((k-numdiv(k)) == t, z++; if(z <= uplim, v263267[z] = k))));
%o A263267 (Sage) # After _David Eppstein_'s Python-code for A088975.
%o A263267 def A263267():
%o A263267 '''Breadth-first reading of irregular tree defined by the edge-relation A049820(child) = parent'''
%o A263267 yield 0
%o A263267 for x in A263267():
%o A263267 for k in [x+1 .. 2*(x+1)]:
%o A263267 if ((k - sloane.A000005(k)) == x): yield k
%o A263267 def take(n,g):
%o A263267 '''Returns a list composed of the next n elements returned by generator g.'''
%o A263267 return [next(g) for _ in range(n)]
%o A263267 take(120, A263267())
%o A263267 (Scheme)
%o A263267 ;; This version creates the list of terms incrementally, using append! function that physically modifies the list at the same time as it is traversed. Otherwise the idea is essentially the same as with Python/Sage-program above:
%o A263267 (define (A263267list_up_to_n_terms_at_least n) (let ((terms-produced (list 0))) (let loop ((startp terms-produced) (endp terms-produced) (k (- n 1))) (cond ((<= k 0) terms-produced) (else (let ((children (children-of-n-in-A049820-tree (car startp)))) (cond ((null? children) (loop (cdr startp) endp k)) (else (begin (append! endp children) (loop (cdr startp) children (- k (length children))))))))))))
%o A263267 (define (children-of-n-in-A049820-tree n) (let loop ((k (A262686 n)) (children (list))) (cond ((<= k n) children) ((= (A049820 k) n) (loop (- k 1) (cons k children))) (else (loop (- k 1) children)))))
%Y A263267 Inverse permutation: A263268.
%Y A263267 Cf. A000005, A049820, A060990, A082284, A155043, A259934, A262508.
%Y A263267 Cf. A262507 (number of terms on row/level n), A263260 (total number of terms in levels 0 .. n).
%Y A263267 Cf. A264988 (the left edge), this differs from A261089 (the least term on each level) for the first time at level 69.
%Y A263267 Cf. A263269 (the right edge).
%Y A263267 Cf. A262686 (maximum term on the level n).
%Y A263267 Cf. A045765 (the leaves of the tree).
%Y A263267 Cf. also permutations A263265 (obtained from this table by sorting each row into ascending order), A263266.
%Y A263267 Cf. also arrays A265751 and A263271.
%Y A263267 Differs from A263265 for the first time at n=31, where a(31) = 40, while A263265(31) = 38.
%Y A263267 Cf. also A088975.
%K A263267 nonn,tabf
%O A263267 0,3
%A A263267 _Antti Karttunen_, Nov 27 2015