OFFSET
1,2
COMMENTS
Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001
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
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
Comments from N. J. A. Sloane, Mar 29 2025 (Start):
Let d(n) = number of 1 <= i <= n such that a(i) < i. The d(i) sequence begins 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, ..., and appears to be A060144 without its initial term.
Let r(n) = 1 if a(n) < a(n+1), otherwise 0, and let f(n) = 1 if a(n) > a(n+1), otherwise 0. Then R = partial sums of r(n) and F = partial sums of f(n) count the rises and falls, respectively, in the present sequence. It appears that R and F are essentially A060143 and A060144 (again).
If a(n) is the k-th term in a monotonically strictly increasing rum of terms, set R(n) = k. It appears that the sequence R(n), n>=1, is essentially A270788.
(End)
REFERENCES
Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
LINKS
Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000
Éric Angelini, Franklin Adams-Watters, Max Alekseyev, A. E. Povolotsky, N. J. A. Sloane, and R. G. Wilson v, a(n) divides the sum of the first a(n) terms of T, Various postings to the old Sequence Fans Mailing List, assembled by N. J. A. Sloane, Dec 24 2024
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 18.
Math Forum, Problem of the Week 818.
Luis Martínez and Iker Malaina, Wythoff-Fibonacci Sequences and a Perturbed Greedy Almost-involution, arXiv:2607.00814 [math.CO], 2026. See p. 2.
Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.
A. Shapovalov, Problem M1517 (in Russian), Kvant 5 (1995), 20-21. English translation appeared in Quantum problem M185, Sept/October 1996 (beware, file is 75Mb).
B. J. Venkatachala, A curious bijection on natural numbers, J. Int. Seq. 12 (2009), Art. 09.8.1.
FORMULA
a(n) = A002251(n-1) + 1. (Corrected by M. F. Hasler, Sep 17 2014)
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
Lim_{n->infinity} max(n,a(n))/min(n,a(n)) = phi = A001622. - Stanislav Sykora, Jun 12 2017
MATHEMATICA
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]
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 *)
Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] - Total[#1]] &, {1}, Range[2, 70]] (* Birkas Gyorgy, May 25 2012 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,nice,changed
AUTHOR
R. K. Guy and Tom Halverson (halverson(AT)macalester.edu)
STATUS
approved