A116159 -id:A116159 - cp's OEIS frontend

A115429 Numbers k such that the concatenation of k with k+8 gives a square.

6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217

Offset: 1

Giovanni Resta, Jan 24 2006

Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 6.
Also numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 8.

			6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.

Cf. A030465, A102567, A115426, A115437, A115428, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115440.
Cf. A116107, A116109, A116113, A116239.
Cf. A116205, A115430, A116335, A116185, A116187, A116317.
Cf. A116131, A116112, A116152, A116159, A116282.

Edited by N. J. A. Sloane, Apr 15 2007

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

Original entry on oeis.org

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

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.

A116160 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 7.

Original entry on oeis.org

4, 94, 210, 294, 994, 5880, 9994, 52888, 99994, 127044, 414180, 999994, 8264470, 9999994, 12456750, 41868508, 99999994, 112670544, 441341880, 468144040, 669421494, 702338994, 715976338, 750005718, 960645294, 999999994

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A020338, A116152, A116159, A116161, A116167, A116291.

cp's OEIS Frontend

A115429 Numbers k such that the concatenation of k with k+8 gives a square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions

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

Views

Author

Keywords

Crossrefs

A116160 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 7.

Views

Author

Keywords

Crossrefs