A116244 Numbers k such that k * (k + 8) is the concatenation of two numbers m and m-7.

94, 461, 532, 714, 818, 994, 3424, 6569, 9994, 90903, 99994, 980198, 999994, 3636357, 6363636, 9999994, 41176464, 58823529, 99999994, 413533834, 426573426, 428571422, 432620005, 567379988, 571428571, 573426567

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

Comments

From Robert Israel, Aug 22 2023: (Start)

Numbers k = a*c-1 such that for some positive integers a,b,c,d,e we have

10^e + 1 = a*b

10^(e-1) <= c*d < 10^e

a*c + 6 = b*d.

Includes 10^k-6 for k >= 2. (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A116113, A116239, A116243, A116245, A116257.

Programs

Maple

F:= proc(d) local R, t,alpha, beta, gamma, delta, B,C,n,m,i0,i,gamma0, delta0;
   R:= NULL;
   t:= 10^d+1;
   for alpha in numtheory:-divisors(t) do
     beta:= t/alpha;
     if igcd(alpha,beta) > 1 then next fi;
     delta0:= 6/beta mod alpha;
     gamma0:= (beta*delta0-6)/alpha;
     B:= 2*alpha*gamma0 + 6;
     C:= gamma0*delta0 - 10^(d-1) - 7;
     if C < 0 then i0:= 0 else i0:= ceil((-B + sqrt(B^2-4*t*C))/(2*t)) fi;
     for i from i0 do
       gamma:= gamma0 + i*beta;
       delta:= delta0 + i*alpha;
       m:= gamma*delta;
       if m -7 >= 10^d then break fi;
       if m - 7 >= 10^(d-1) then R:= R, alpha*gamma-1 fi;
     od
   od;
   sort(convert({R},list))
end proc:
seq(op(F(d)),d=1..10); # Robert Israel, Aug 22 2023

Mathematica

a[n_] := Module[{solutions = {}, kvalues, e = 2}, While[Length[solutions] < n, sol = Solve[{a*b == 10^e + 1, 10^(e - 1) <= c*d < 10^e, a*c + 6 == b*d, a > 0, b > 0, c > 0, d > 0}, {a, b, c, d}, Integers]; kvalues = (a*c - 1) /. sol; solutions = Union[solutions, kvalues]; e++]; Take[solutions, n]]; a[26] (* Robert P. P. McKone, Aug 22 2023 *)