A116289 -id:A116289 - cp's OEIS frontend

A116288 Numbers k such that k*(k+4) gives the concatenation of a number m with itself.

7, 97, 451, 546, 689, 854, 997, 4380, 5617, 9997, 72728, 99997, 346531, 653466, 999997, 9090906, 9999997, 94117644, 99999997, 334665331, 336032385, 378253326, 390977440, 439928489, 483516484, 516483513, 560071508

Offset: 1

Giovanni Resta, Feb 06 2006

Contains 10^m - 3 hence the sequence is infinite. - David A. Corneth, Feb 20 2024

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A116125, A116275, A116287, A116289, A116297.

from itertools import count, islice
from sympy import sqrt_mod
def A116288_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(4,a,all_roots=True)):
            if a*b <= k**2-4 < a*(a-1):
                yield k-2
A116288_list = list(islice(A116288_gen(),40)) # Chai Wah Wu, Feb 20 2024

A116290 Numbers k such that k * (k+6) is the concatenation of a number m with itself.

Original entry on oeis.org

5, 95, 462, 533, 715, 819, 995, 3425, 6570, 9995, 90904, 99995, 980199, 999995, 3636358, 6363637, 9999995, 41176465, 58823530, 99999995, 413533835, 426573427, 428571423, 432620006, 567379989, 571428572, 573426568, 586466160, 686261108, 725274726, 727272722

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 08 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-5 for all d >= 1. (End)

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

Cf. A116159, A116282, A116289, A116291, A116304.

f:= proc(d) 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,-6],[a,b]) by t do
       q:= x*(x+6)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort([R]);
end proc:
seq(op(f(d)),d=1..10); # Robert Israel, Apr 08 2025

Name edited by Robert Israel, Apr 08 2025

A116158 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 5.

Original entry on oeis.org

6, 96, 150, 186, 324, 376, 666, 996, 2046, 3000, 9996, 82650, 99996, 100384, 466716, 999996, 1322316, 4049584, 9999996, 67820074, 99999996, 110003884, 135734074, 156502836, 196043286, 213017754, 238849000, 261405396

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116112, A116125, A116159, A116172, A116289.

A116281 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

39, 57, 32262231, 67737765, 79321055, 3341093417798787499092, 3861488851737861033960, 4747922651210186579786, 5252077348789813420210, 6138511148262138966036, 6658906582201212500904, 7232275368591793618230

Offset: 1

Giovanni Resta, Feb 06 2006

			79321055 * 79321060 = 62918301//62918300, where // denotes concatenation.

Cf. A028557, A116112, A116276, A116280, A116282, A116289.

A116303 n times n+5 gives the concatenation of two numbers m and m+2.

Original entry on oeis.org

3, 84, 76981, 714688, 952312, 90438189, 96320542, 32980078899027, 34346653774236, 42816188292271, 42881990066486, 57118009933510, 57183811707725, 65653346225760, 67019921100969, 81321742742208

Offset: 1

Giovanni Resta, Feb 06 2006

			96320542 * 96320547 = 92776472//92776474, where // denotes
concatenation.

Cf. A116172, A116289, A116302, A116304, A116310.

cp's OEIS Frontend

A116288 Numbers k such that k*(k+4) gives the concatenation of a number m with itself.

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A116290 Numbers k such that k * (k+6) is the concatenation of a number m with itself.

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A116158 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 5.

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A116281 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-1.

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A116303 n times n+5 gives the concatenation of two numbers m and m+2.

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