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A116300 n times n+9 gives the concatenation of two numbers m and m+1.

%I A116300 #10 May 07 2024 14:49:44
%S A116300 26,66,3416102887775247376839416334668635,
%T A116300 3756559953325598880263233435801764,4313503800489302411917772257282208
%N A116300 n times n+9 gives the concatenation of two numbers m and m+1.
%H A116300 Robert Israel, <a href="/A116300/b116300.txt">Table of n, a(n) for n = 1..28</a>
%e A116300 66 * 75 = 49//50, where // denotes concatenation.
%p A116300 F:= proc(d) local t, g, Cands:
%p A116300   t:= 10^d+1;
%p A116300   if NumberTheory:-QuadraticResidue(85,t) <> 1 then return NULL fi;
%p A116300   Cands:= map(s -> rhs(op(s)), [msolve(x^2 + 9*x - 1, t)]);
%p A116300   g:= proc(r) local v; v:= r^2 + 9*r - 1; v >= t*(t-11)/10 and v < t*(t-2) end proc;
%p A116300   op(sort(select(g, Cands)));
%p A116300 end proc:
%p A116300 map
%p A116300 map(F, [$1..82]); # _Robert Israel_, Aug 25 2023
%o A116300 (Python)
%o A116300 from itertools import count, islice
%o A116300 from sympy import sqrt_mod_iter
%o A116300 def A116300_gen(): # generator of terms
%o A116300     for l in count(1):
%o A116300         m = 10**l+1
%o A116300         k, r, dlist = m*(m-11)/10, m*(m-2), []
%o A116300         for a in sqrt_mod_iter(85,m):
%o A116300             d = ((a if a&1 else a+m)>>1)-4
%o A116300             if k<d*(d+9)<=r:
%o A116300                 dlist.append(d)
%o A116300         yield from sorted(dlist)
%o A116300 A116300_list = list(islice(A116300_gen(),14)) # _Chai Wah Wu_, May 07 2024
%Y A116300 Cf. A116169, A116293, A116299, A116306.
%K A116300 nonn,base
%O A116300 1,1
%A A116300 _Giovanni Resta_, Feb 06 2006