A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

363, 637, 714, 923, 8905, 81818, 336633, 663367, 7272727, 76470589, 333666333, 405436668, 428571429, 447710185, 454545454, 473684211, 526315789, 545454546, 552289815, 571428571, 594563332, 666333667, 692307693, 711446449, 762237762, 834008097, 859982123, 879120879, 902255640, 974025975, 980861244

Offset

1,1

Comments

From Robert Israel, Apr 08 2025: (Start)

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

Includes (10^(3*d)-1)/3 + (10^d-1)*10^d/3 and 2*(10^(3*d)-1)/3 - (10^d-1)*10^d/3 + 1 for all d >= 1. (End)

Links

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

Formula

A161356(n) = a(n)*(a(n)+1). - Michael S. Branicky, Jul 11 2025

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, 1)), d=1..10); # Robert Israel, Apr 08 2025

Crossrefs

Cf. A116154, A116272, A116286, A116294, A161356.

Sequence in context: A098253 A031517 A192449 * A031697 A158309 A098251

Adjacent sequences: A116282 A116283 A116284 * A116286 A116287 A116288

Keyword

nonn,base

Author

Giovanni Resta, Feb 06 2006

Extensions

More terms from Robert Israel, Apr 08 2025

Status

approved