A116291 Numbers k such that k * (k + 7) is the concatenation of a number m with itself.

4, 94, 455, 539, 994, 7665, 9994, 72721, 99994, 356429, 643565, 999994, 9090910, 9999994, 35294118, 64705876, 99999994, 335664329, 664335665, 684210520, 818181819, 838056674, 846153847, 866028702, 980125139, 999999994

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

Comments

From Robert Israel, Apr 09 2025: (Start)

Numbers k such that k * (k + 7) = (10^d + 1) * m for some d and m where m has d digits.

Contains 10^d - 6 for all d >= 1. (End)

Links

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

Crossrefs

Cf. A116160, A116283, A116290, A116292, A116298.

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:
[seq(op(q(d,7)),d=1..10)]; # Robert Israel, Apr 09 2025