A116293 -id:A116293 - cp's OEIS frontend

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

Giovanni Resta, Feb 06 2006

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)

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

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

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

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 *)

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

A116300 n times n+9 gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

26, 66, 3416102887775247376839416334668635, 3756559953325598880263233435801764, 4313503800489302411917772257282208

Offset: 1

Giovanni Resta, Feb 06 2006

			66 * 75 = 49//50, where // denotes concatenation.

Robert Israel, Table of n, a(n) for n = 1..28

Cf. A116169, A116293, A116299, A116306.

F:= proc(d) local t, g, Cands:
  t:= 10^d+1;
  if NumberTheory:-QuadraticResidue(85,t) <> 1 then return NULL fi;
  Cands:= map(s -> rhs(op(s)), [msolve(x^2 + 9*x - 1, t)]);
  g:= proc(r) local v; v:= r^2 + 9*r - 1; v >= t*(t-11)/10 and v < t*(t-2) end proc;
  op(sort(select(g, Cands)));
end proc:
map
map(F, [$1..82]); # Robert Israel, Aug 25 2023

from itertools import count, islice
from sympy import sqrt_mod_iter
def A116300_gen(): # generator of terms
    for l in count(1):
        m = 10**l+1
        k, r, dlist = m*(m-11)/10, m*(m-2), []
        for a in sqrt_mod_iter(85,m):
            d = ((a if a&1 else a+m)>>1)-4
            if kA116300_list = list(islice(A116300_gen(),14)) # Chai Wah Wu, May 07 2024

A116162 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 9.

Original entry on oeis.org

2, 92, 180, 332, 486, 536, 992, 9720, 9992, 13220, 40502, 99992, 941486, 999992, 2066120, 2975202, 9999992, 77854680, 99999992, 123250136, 162423540, 243964010, 256109750, 356387462, 421110162, 542936282, 547361900

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116153, A116161, A116169, A116293.

A116284 n times n+9 gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

32, 60, 804, 4778, 5214, 7406, 48208886, 51791106, 77620651, 4291816088, 4482541430, 5517458562, 5708183904, 9036996876, 9737360643, 3579205562000136225339579331, 3939042613182345526485238179

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116153, A116278, A116283, A116293.

cp's OEIS Frontend

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

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A116300 n times n+9 gives the concatenation of two numbers m and m+1.

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A116162 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 9.

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A116284 n times n+9 gives the concatenation of two numbers m and m-1.

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