A116304 -id:A116304 - cp's OEIS frontend

A116173 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 6.

6752089, 6448802889351008245, 18894512461523256139943105859903480218905, 31958875438439894736354375209245786214798

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 4.

If k+2 and k-5 have the same number of digits, the k is also in A116126, because k//k+2 = 10^d*k + k + 2 = m*(m+6) then implies a representation k//k-5 = 10^d*k + k - 5 = m*(m+6)-7 = (m-1)*(m+7). - R. J. Mathar, Aug 10 2008

			6752089//6752091 = 8217107 * 8217113, where // denotes concatenation.
6752089//6752096 = 8217108 * 8217112.

Cf. A116159, A116172, A116174, A116187, A116304.

Cf. A115430, A116205, A116207, A116187, A116337.

Edited by N. J. A. Sloane, Apr 12 2007

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

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.

A116305 Numbers k such that k*(k+8) gives the concatenation of two numbers m and m+2.

Original entry on oeis.org

3377, 6616, 7027, 49941473, 50058520, 97000297, 785321170880, 799019801017, 989301366860, 997000002997, 4044997109223941, 4136014586911892, 4248535260473901, 4366325506691845, 4457342984379796, 5542657015620197

Offset: 1

Giovanni Resta, Feb 06 2006

			97000297 * 97000305 = 94090583//94090585, where // denotes concatenation.

Cf. A116174, A116299, A116304, A116306, A116312.

A116318 n times n+6 gives the concatenation of two numbers m and m+4.

Original entry on oeis.org

32, 63, 45211540, 54788455, 78317867, 72388446037855609175404257, 73247326752475247326752672, 75267524732673267524732472, 76126405447292905676080887, 42053270820132695893502981749303

Offset: 1

Giovanni Resta, Feb 06 2006

			78317867 * 78317873 = 61336887//61336891, where // denotes
concatenation.

Cf. A116187, A116304, A116317, A116319, A116325.

A116337 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m+7.

Original entry on oeis.org

8217108, 8030443878983905982, 43467818511541680701794365325328847002052, 56532181488458319298205634674671152997945

Offset: 1

Giovanni Resta, Feb 06 2006

			8217108 * 8217112 = 6752089//6752096, where // denotes concatenation.

Cf. A116173, A116323, A116336, A116338, A116350.

Cf. A116304, A116257.

cp's OEIS Frontend

A116173 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 6.

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

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A116305 Numbers k such that k*(k+8) gives the concatenation of two numbers m and m+2.

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A116318 n times n+6 gives the concatenation of two numbers m and m+4.

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A116337 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m+7.

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