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A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.

%I A082607 #17 Aug 25 2025 11:33:57
%S A082607 1,2,5,10,17,26,37,50,65,34,13,25,41,61,85,113,145,122,101,82,293,634,
%T A082607 1105,53,109,185,74,149,250,377,205,146,97,58,29,73,137,221,181,650,
%U A082607 541,442,353,274,953,2042,3541,5450,409,173,370,289,218,157,106,337,698
%N A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.
%C A082607 Conjecture: this is a permutation of A008784. - _Robert Israel_, Aug 25 2025
%H A082607 Robert Israel, <a href="/A082607/b082607.txt">Table of n, a(n) for n = 0..10000</a>
%p A082607 N:= 10000: # for terms before the first term > N
%p A082607 Cands:= select(t -> numtheory:-quadres(-1,t) = 1, [$2..N]): nc:= nops(Cands):
%p A082607 R:= 1: r:= 1:
%p A082607 do
%p A082607   found:= false;
%p A082607   for i from 1 to nc do
%p A082607     if issqr(r*Cands[i]-1) then
%p A082607        found:= true;
%p A082607        r:= Cands[i];
%p A082607        R:= R,r;
%p A082607        Cands:= subsop(i=NULL,Cands);
%p A082607        nc:= nc-1;
%p A082607        break
%p A082607     fi
%p A082607   od;
%p A082607   if not found then break fi
%p A082607 od:
%p A082607 R; # _Robert Israel_, Aug 25 2025
%t A082607 l = {1}; Do[k = 1; While[MemberQ[l, k] || !IntegerQ[Sqrt[k*Last[l]-1]], k++ ]; AppendTo[l, k], {n, 50}]; l (* _Ryan Propper_, Jun 13 2006 *)
%o A082607 (PARI) a=[1];print1(1",");for(n=2,100,k=1;f=1;while(f,if(issquare(k*a[n-1]-1),f=0;for(i=1,n-1,if(a[i]==k,f=1)));k++);a=concat(a,k-1);print1(k-1",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
%Y A082607 Contained in A008784. Cf. A082608, A082609, A082610, A082611, A082612.
%K A082607 nonn,changed
%O A082607 0,2
%A A082607 _Amarnath Murthy_, Apr 28 2003
%E A082607 Corrected and extended by _Ryan Propper_, Jun 13 2006
%E A082607 Definition corrected by _R. J. Mathar_, Nov 12 2006
%E A082607 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007