A125717 a(0)=0; thereafter a(n) = the smallest nonnegative integer not already in the sequence such that a(n-1) is congruent to a(n) (mod n).

0, 1, 3, 6, 2, 7, 13, 20, 4, 22, 12, 23, 11, 24, 10, 25, 9, 26, 8, 27, 47, 5, 49, 72, 48, 73, 21, 75, 19, 77, 17, 79, 15, 81, 115, 45, 117, 43, 119, 41, 121, 39, 123, 37, 125, 35, 127, 33, 129, 31, 131, 29, 133, 80, 134, 189, 245, 74, 16, 193, 253, 70, 132, 69, 197, 67, 199, 65

Offset: 0

Author

Leroy Quet, Feb 01 2007

Keywords

• nonn
• nice
• look

Comments

This sequence seems likely to be a permutation of the nonnegative integers.

A245340(n) = smallest m such that a(m) = n, or -1 if n never appears.

See A245394 and A245395 for record values of a(n) and where they occur. - Reinhard Zumkeller, Jul 21 2014

See A370956 and A370959 for record values of the inverse A245340 and where they occur. - N. J. A. Sloane, Apr 29 2024

A very nice (maybe the most natural) variant of Recamán's sequence A005132. - M. F. Hasler, Nov 03 2014

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..100000 (first 10000 terms from Ferenc Adorján)
• Ferenc Adorján, Some characteristics of Leroy Quet's permutation sequences
• N. J. A. Sloane, Log-log plot of A370956 vs A370959 (shows terms in A125717 that take the longest to appear).
• Index entries for sequences related to Recamán's sequence
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A245340 (inverse), A370957 (first differences), A245394 & A245395 (records in this sequence), A370956 & A370959 (records in inverse).

See also A005132 (Recaman), A099506, A125715, A125718, A125725.

Programs

Haskell

import Data.IntMap (singleton, member, (!), insert)
a125717 n = a125717_list !! n
a125717_list =  0 : f [1..] 0 (singleton 0 0) where
   f (v:vs) w m = g (reverse[w-v,w-2*v..1] ++ [w+v,w+2*v..]) where
     g (x:xs) = if x `member` m then g xs else x : f vs x (insert x v m)
-- Reinhard Zumkeller, Jul 21 2014

Mathematica

f[l_List] := Block[{n = Length[l], k = Mod[l[[ -1]], n]},While[MemberQ[l, k], k += n];Append[l, k]];Nest[f, {0}, 70] (* Ray Chandler, Feb 04 2007, updated for change to offset Oct 10 2019 *)

PARI

{Quet_p2(n)=/* Permutation sequence a'la Leroy Quet, A125717 */local(x=[1],k=0,w=1); for(i=2,n,if((k=x[i-1]%i)==0,k=i);while(bittest(w,k-1)>0,k+=i);x=concat(x,k);w+=2^(k-1)); return(x)} \\ Ferenc Adorjan

PARI

A125717(n,show=0)={my(u=1,a);for(n=1,n,a%=n;while(bittest(u,a),a+=n);u+=1<M. F. Hasler, Nov 03 2014

Python

from itertools import count, islice
def agen(): # generator of terms
    an, aset = 0, {0}
    for n in count(1):
        yield an
        an = next(m for m in count(an%n, n) if m not in aset)
        aset.add(an)
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 07 2023

Extensions

Extended by Ray Chandler, Feb 04 2007

a(0) added by Franklin T. Adams-Watters, Mar 31 2014

Edited by N. J. A. Sloane, Mar 15 2024