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%I #86 Feb 12 2024 06:38:00
%S 1,3,2,6,8,4,11,5,14,16,7,19,21,9,24,10,27,29,12,32,13,35,37,15,40,42,
%T 17,45,18,48,50,20,53,55,22,58,23,61,63,25,66,26,69,71,28,74,76,30,79,
%U 31,82,84,33,87,34,90,92,36,95,97,38,100,39,103,105,41,108,110,43,113
%N a_1, a_2, ..., is a permutation of the positive integers such that the average of each initial segment is an integer, using the greedy algorithm to define a_n.
%C Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - _Howard A. Landman_, Sep 25 2001
%C That each initial segment has an integer average is trivially equivalent to the sum of the first n elements always being divisible by n. - _Franklin T. Adams-Watters_, Jul 07 2014
%C Also, a lexicographically minimal sequence of distinct positive integers such that all values of a(n)-n are also distinct. - _Ivan Neretin_, Apr 18 2015
%D Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
%H Franklin T. Adams-Watters, <a href="/A019444/b019444.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Shallit, <a href="https://arxiv.org/abs/2308.06544">Proving properties of some greedily-defined integer recurrences via automata theory</a>, arXiv:2308.06544 [cs.DM], August 12 2023.
%H A. Shapovalov, <a href="http://kvant.mccme.ru/1995/05/zadachnik_kvanta_matematika.htm">Problem M1517</a> (in Russian), Kvant 5 (1995), 20-21. English translation appeared in <a href="http://static.nsta.org/pdfs/QuantumV7N1.pdf">Quantum problem M185</a>, Sept/October 1996 (beware, file is 75Mb).
%H The Math Forum, <a href="http://mathforum.org/wagon/fall96/p818.html">Problem of the Week 818</a>
%H B. J. Venkatachala, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Venkatachala/venkatachala2.html">A curious bijection on natural numbers</a>, JIS 12 (2009) 09.8.1.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A002251(n-1) + 1. (Corrected by _M. F. Hasler_, Sep 17 2014)
%F Let s(n) = (1/n)*Sum_{k=1..n} a(k) = A019446(n). Then if s(n-1) does not occur in a(1),...,a(n-1), a(n) = s(n) = s(n-1); otherwise, a(n) = s(n-1) + n and s(n) = s(n-1) + 1. - _Franklin T. Adams-Watters_, May 20 2010
%F Lim_{n->infinity} max(n,a(n))/min(n,a(n)) = phi = A001622. - _Stanislav Sykora_, Jun 12 2017
%t a[1]=1; a[n_] := a[n]=Module[{s, v}, s=a/@Range[n-1]; For[v=Mod[ -Plus@@s, n], v<1||MemberQ[s, v], v+=n, Null]; v]
%t lst = {1}; f[s_List] := Block[{k = 1, len = 1 + Length@ lst, t = Plus @@ lst}, While[ MemberQ[s, k] || Mod[k + t, len] != 0, k++ ]; AppendTo[lst, k]]; Nest[f, lst, 69] (* _Robert G. Wilson v_, May 17 2010 *)
%t Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] - Total[#1]] &, {1}, Range[2, 70]] (* _Birkas Gyorgy_, May 25 2012 *)
%o (PARI) al(n)=local(v,s,fnd);v=vector(n);v[1]=s=1;for(k=2,n,fnd=0;for(i=1,k-1,if(v[i]==s,fnd=1;break));v[k]=if(fnd,s+k,s);s+=fnd);v \\ _Franklin T. Adams-Watters_, May 20 2010
%o (PARI) A019444_upto(N, c=0, A=Vec(1, N))={for(n=2, N, A[n]||(#A<A[n]=n+c++)|| A[n+c]=n); A} \\ _M. F. Hasler_, Nov 27 2019
%Y Cf. A019445, A019446, A243700, A001622.
%K nonn,nice
%O 1,2
%A _R. K. Guy_ and Tom Halverson (halverson(AT)macalester.edu)
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