A116292 Numbers k such that k * (k + 8) is the concatenation of a number m with itself.

3, 93, 377, 616, 707, 902, 993, 8760, 9993, 45455, 54538, 99993, 693062, 999993, 8181811, 9999993, 88235287, 99999993, 327935223, 330669331, 363636364, 418318516, 428571429, 461538454, 538461539, 571428564, 581681477, 636363629, 669330662, 672064770, 691571587, 756506652, 781954880, 789473685

Offset

1,1

Comments

From Robert Israel, Apr 09 2025: (Start)

Numbers k such that k * (k + 6) = (10^d + 1) * m for some d and m where m has d digits.

Contains 10^d-7 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:
seq(op(q(d, 8)), d=1..10); # Robert Israel, Apr 09 2025

Mathematica

ccnQ[n_]:=With[{ccc=With[{c=n(n+8)}, TakeDrop[IntegerDigits[c], IntegerLength[c]/2]]}, ccc[[1]]==ccc[[2]]];  Select[Range[10^6], ccnQ]//Quiet (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Jul 05 2025 *)

Crossrefs

Cf. A116161, A116277, A116291, A116293, A116299.

Sequence in context: A116161 A285004 A288070 * A139543 A241982 A037113

Adjacent sequences: A116289 A116290 A116291 * A116293 A116294 A116295

Keyword

nonn,base

Author

Giovanni Resta, Feb 06 2006

Extensions

Name edited and more terms from Robert Israel, Apr 09 2025

Status

approved