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