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

26, 66, 3416102887775247376839416334668635, 3756559953325598880263233435801764, 4313503800489302411917772257282208

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

Examples

			66 * 75 = 49//50, where // denotes concatenation.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..28

Crossrefs

Cf. A116169, A116293, A116299, A116306.

Programs

Maple

F:= proc(d) local t, g, Cands:
  t:= 10^d+1;
  if NumberTheory:-QuadraticResidue(85,t) <> 1 then return NULL fi;
  Cands:= map(s -> rhs(op(s)), [msolve(x^2 + 9*x - 1, t)]);
  g:= proc(r) local v; v:= r^2 + 9*r - 1; v >= t*(t-11)/10 and v < t*(t-2) end proc;
  op(sort(select(g, Cands)));
end proc:
map
map(F, [$1..82]); # Robert Israel, Aug 25 2023

Python

from itertools import count, islice
from sympy import sqrt_mod_iter
def A116300_gen(): # generator of terms
    for l in count(1):
        m = 10**l+1
        k, r, dlist = m*(m-11)/10, m*(m-2), []
        for a in sqrt_mod_iter(85,m):
            d = ((a if a&1 else a+m)>>1)-4
            if kA116300_list = list(islice(A116300_gen(),14)) # Chai Wah Wu, May 07 2024