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

1,1

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

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

Crossrefs

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

Sequence in context: A284691 A147829 A116274 * A306449 A160946 A359078

Adjacent sequences: A116283 A116284 A116285 * A116287 A116288 A116289

Keyword

nonn,base

Author

Giovanni Resta, Feb 06 2006

Extensions

More terms from Robert Israel, Apr 08 2025

Status

approved