A116272 -id:A116272 - cp's OEIS frontend

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

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

A116265 n times n+1 gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

96013, 3812029082958589067221817215314, 6187970917041410932778182784686, 3189731673290266386838927279556315882

Offset: 1

Giovanni Resta, Feb 06 2006

			96013 * 96014 = 92185//92182, where // denotes concatenation.

Cf. A116134, A116251, A116266, A116272.

A116273 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

90, 9078, 9900, 733673, 999000, 88225294, 99990000, 8900869207, 9296908811, 9604060396, 9999900000, 326666333266, 673333666733, 700730927007, 972603739726, 999999000000, 34519562953736, 39737862788837

Offset: 1

Giovanni Resta, Feb 06 2006

			90 is a member since 90*92 = 8280 = 82 80.
9078 is a member since 9078*9080 = 82428240 = 8242 8240.

Cf. A054214, A116266, A116272, A116274, A116279.

A116141 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 1.

Original entry on oeis.org

4, 5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382

Offset: 1

Giovanni Resta, Feb 06 2006

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

Cf. A116134, A054214, A116154, A116272.

cp's OEIS Frontend

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

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A116265 n times n+1 gives the concatenation of two numbers m and m-3.

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A116273 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-2.

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A116141 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 1.

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