A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

9, 99, 427, 572, 726, 845, 999, 7809, 9999, 36364, 63635, 99999, 326733, 673266, 999999, 4545453, 5454546, 9999999, 47058822, 52941177, 99999999, 331983806, 332667333, 384615385, 422892897, 475524476, 524475523, 577107102, 615384614, 667332666, 668016193, 719964245, 758241757, 804511279

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

Comments

From Robert Israel, Apr 08 2025: (Start)

Numbers k such that k * (k + 2) = (10^d + 1) * m for some d and m where m has d digits.

Contains 10^d - 1 for all d >= 1. (End)

Links

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

Crossrefs

Cf. A116136, A116279, A116285, A116287, A116295.

Programs

Maple

q:= proc(d,m) local R,t,a,b,x,q;
   t:= 10^d+1;
   R:= NULL;
   for a in numtheory:-divisors(t) do
     b:= t/a;
     if igcd(a,b) > 1 then next fi;
     for x from chrem([0,-m],[a,b]) by t do
       q:= x*(x+m)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort(convert({R},list));
end proc:
A:=[seq(op(q(d,2)),d=1..10)]; # Robert Israel, Apr 08 2025

Extensions

More terms from Robert Israel, Apr 08 2025