A116285 -id:A116285 - cp's OEIS frontend

A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

9, 99, 427, 572, 726, 845, 999, 7809, 9999, 36364, 63635, 99999, 326733, 673266, 999999, 4545453, 5454546, 9999999, 47058822, 52941177, 99999999, 331983806, 332667333, 384615385, 422892897, 475524476, 524475523, 577107102, 615384614, 667332666, 668016193, 719964245, 758241757, 804511279

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 08 2025: (Start)
Numbers k such that k * (k + 2) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d - 1 for all d >= 1. (End)

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

Cf. A116136, A116279, A116285, A116287, A116295.

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:
A:=[seq(op(q(d,2)),d=1..10)]; # Robert Israel, Apr 08 2025

More terms from Robert Israel, Apr 08 2025

A116294 Numbers k such that k*(k+1) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

3, 7, 78, 80919, 91809, 326510, 475025, 524975, 673490, 4323777, 4767132, 5232868, 5676223, 4083911141, 4975000250, 5024999750, 5916088859, 9503960496, 9604950396, 4186904462792, 4313465946775, 5686534053225, 5813095537208, 7504715871407, 7631277355390

Offset: 1

Giovanni Resta, Feb 06 2006

			80919 * (80919 + 1) = 6547965480, the concatenation of 65479 and 65479 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..2000

Cf. A116163, A116285, A116295, A116301.

Union @@ ((x /. List@ ToRules@ Reduce[x (x+1) == 10^# y +y+1 && x>0 && 10^(#-1) <= y+1 < 10^#, {x,y}, Integers]) & /@ Range[13] /. x->{}) (* Giovanni Resta, Jul 08 2018 *)

More terms from Giovanni Resta, Jul 08 2018

A116154 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 1.

Original entry on oeis.org

132, 406, 510, 852, 7930, 66942, 113322, 440056, 5289256, 58477510, 111333222, 164378892, 183673470, 200444410, 206611570, 224376732, 277008310, 297520662, 305024040, 326530612, 353505556, 444000556, 479289942

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116141, A116136, A116163, A116285.

A116272 n times n+1 gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

6, 7282818, 8171728, 4550754022124826467, 5449245977875173533, 8176518705147900806, 8951999555715882882349355522889, 9861090464806791973258446431980

Offset: 1

Giovanni Resta, Feb 06 2006

			8171728 * 8171729 = 6677714//6677712, where // denotes
concatenation.

Cf. A116141, A116265, A116273, A116285.

A161356 Pronic numbers A002378 (AA) that can be divided into two equal strings (A).

Original entry on oeis.org

132132, 406406, 510510, 852852, 79307930, 6694266942, 113322113322, 440056440056, 52892565289256, 5847751058477510, 111333222111333222, 164378892164378892, 183673470183673470, 200444410200444410, 206611570206611570, 224376732224376732, 277008310277008310, 297520662297520662

Offset: 1

Claudio Meller, Jun 07 2009

			132132 = 363 x 364 and can be divided into two equal strings (132), so it is a term.
406406 = 637 x 638, 510510 = 714 x 715, 852852 = 923 x 924, 79307930 = 8905 x 8906 are terms.

Cf. A116285.

a(n) = A116285(n)*(A116285(n)+1). - Michael S. Branicky, Jul 11 2025

a(6)-a(14) from Donovan Johnson, Jun 15 2009

a(15)-a(18) from Michael S. Branicky, Jul 11 2025 using A116285

cp's OEIS Frontend

A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

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A116294 Numbers k such that k*(k+1) gives the concatenation of two numbers m and m+1.

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A116154 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 1.

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A116272 n times n+1 gives the concatenation of two numbers m and m-2.

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A161356 Pronic numbers A002378 (AA) that can be divided into two equal strings (A).

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