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

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base

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

Crossrefs

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

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,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 *)

Extensions

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