A125717 - OEIS

%I #72 May 01 2024 19:28:55

%S 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,

%T 21,75,19,77,17,79,15,81,115,45,117,43,119,41,121,39,123,37,125,35,

%U 127,33,129,31,131,29,133,80,134,189,245,74,16,193,253,70,132,69,197,67,199,65

%N 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).

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

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

%C See A245394 and A245395 for record values of a(n) and where they occur. - _Reinhard Zumkeller_, Jul 21 2014

%C See A370956 and A370959 for record values of the inverse A245340 and where they occur. - _N. J. A. Sloane_, Apr 29 2024

%C A very nice (maybe the most natural) variant of Recamán's sequence A005132. - _M. F. Hasler_, Nov 03 2014

%H Reinhard Zumkeller, <a href="/A125717/b125717.txt">Table of n, a(n) for n = 0..100000</a> (first 10000 terms from Ferenc Adorján)

%H Ferenc Adorján, <a href="https://web.archive.org/web/20210629063154/http://web.t-online.hu/fadorjan/l_quet.pdf">Some characteristics of Leroy Quet's permutation sequences</a>

%H N. J. A. Sloane, <a href="/A125717/a125717.jpg">Log-log plot of A370956 vs A370959</a> (shows terms in A125717 that take the longest to appear).

%H <a href="/index/Rea#Recaman">Index entries for sequences related to Recamán's sequence</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t 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 *)

%o (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]

%o (Haskell)

%o import Data.IntMap (singleton, member, (!), insert)

%o a125717 n = a125717_list !! n

%o a125717_list =  0 : f [1..] 0 (singleton 0 0) where

%o    f (v:vs) w m = g (reverse[w-v,w-2*v..1] ++ [w+v,w+2*v..]) where

%o      g (x:xs) = if x `member` m then g xs else x : f vs x (insert x v m)

%o -- _Reinhard Zumkeller_, Jul 21 2014

%o (PARI) A125717(n,show=0)={my(u=1,a);for(n=1,n,a%=n;while(bittest(u,a),a+=n);u+=1<<a;show&&print1(a","));a} \\ _M. F. Hasler_, Nov 03 2014

%o (Python)

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     an, aset = 0, {0}

%o     for n in count(1):

%o         yield an

%o         an = next(m for m in count(an%n, n) if m not in aset)

%o         aset.add(an)

%o print(list(islice(agen(), 70))) # _Michael S. Branicky_, Jun 07 2023

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

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

%K nonn,nice,look

%O 0,3

%A _Leroy Quet_, Feb 01 2007

%E Extended by _Ray Chandler_, Feb 04 2007

%E a(0) added by _Franklin T. Adams-Watters_, Mar 31 2014

%E Edited by _N. J. A. Sloane_, Mar 15 2024