A116294 -id:A116294 - cp's OEIS frontend

A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

1, 5, 61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471, 5632076031055, 5823639407489

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 3.

			1 is a member since 12 = 3*4; also 10 = 2*5.
5 is a member since 56 = 7*8; also 54 = 6*9.

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

Cf. A116154, A115426, A116170, A116294.

Cf. A116143, A102567, A116112, A116130, A116280.

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

Edited by N. J. A. Sloane, Apr 15 2007, Jun 27 2009

More terms from Giovanni Resta, Jul 08 2018

A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

Original entry on oeis.org

363, 637, 714, 923, 8905, 81818, 336633, 663367, 7272727, 76470589, 333666333, 405436668, 428571429, 447710185, 454545454, 473684211, 526315789, 545454546, 552289815, 571428571, 594563332, 666333667, 692307693, 711446449, 762237762, 834008097, 859982123, 879120879, 902255640, 974025975, 980861244

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 08 2025: (Start)
Numbers k such that k * (k + 1) = (10^d + 1) * m for some d and m where m has d digits.
Includes (10^(3*d)-1)/3 + (10^d-1)*10^d/3 and 2*(10^(3*d)-1)/3 - (10^d-1)*10^d/3 + 1 for all d >= 1. (End)

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

Cf. A116154, A116272, A116286, A116294, A161356.

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,1)),d=1..10); # Robert Israel, Apr 08 2025

A161356(n) = a(n)*(a(n)+1). - Michael S. Branicky, Jul 11 2025

More terms from Robert Israel, Apr 08 2025

A116295 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

8873, 9010, 83352841, 99000100, 329767122287, 670232877712, 738226276372, 933006600340, 999000001000, 3779410975143114, 3872816717528066, 4250291784692549, 4278630943941866, 4372036686326818, 4749511753491301

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Jun 06 2018: (Start)
Numbers k such that 10^m+1 | (k+1)^2-2 where (k+1)^2 has 2*m digits.
Includes 10^i - 10^(3*i) + 10^(4*i) for all i >= 1. (End)

			99000100 * 99000102 = 98010199//98010200, where // denotes concatenation.

Robert Israel, Table of n, a(n) for n = 1..600

Cf. A115426, A116286, A116294, A116296, A116308.

Res:= NULL:
for d from 1 to 40 do
  Res:= Res, op(sort(select(t -> t^2 >= 10^(2*d-1),map(t -> rhs(op(t))-1,[msolve(x^2=2, 10^d+1)]))))
od:
Res; # Robert Israel, Jun 06 2018

A116301 n times n+1 gives the concatenation of two numbers m and m+2.

Original entry on oeis.org

768, 859, 911, 3286, 6714, 45453, 54547, 990101, 8181820, 70588234, 343130555, 362637364, 363636362, 420053632, 421052633, 497975710, 502024290, 578947367, 579946368, 636363638, 637362636, 656869445, 706766919

Offset: 1

Giovanni Resta, Feb 06 2006

Robert Israel, Table of n, a(n) for n = 1..9000

Cf. A116170, A116294, A116302, A116307.

As:= {}:
for m from 2 to 20 do
   acands:= map(t -> rhs(op(t)), [msolve(a*(a+1)=2, 10^m+1)]);
   bcands:= map(t -> t*(t+1) mod 10^m, acands);
   good:= select(t -> bcands[t]>=10^(m-1), [$1..nops(acands)]);
   As:= As union convert(acands[good],set);
od:
sort(convert(As,list)); # Robert Israel, Aug 20 2019

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A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

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A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

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A116295 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+1.

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A116301 n times n+1 gives the concatenation of two numbers m and m+2.

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