A116292 -id:A116292 - cp's OEIS frontend

A116291 Numbers k such that k * (k + 7) is the concatenation of a number m with itself.

4, 94, 455, 539, 994, 7665, 9994, 72721, 99994, 356429, 643565, 999994, 9090910, 9999994, 35294118, 64705876, 99999994, 335664329, 664335665, 684210520, 818181819, 838056674, 846153847, 866028702, 980125139, 999999994

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 09 2025: (Start)
Numbers k such that k * (k + 7) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d - 6 for all d >= 1. (End)

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

Cf. A116160, A116283, A116290, A116292, A116298.

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,7)),d=1..10)]; # Robert Israel, Apr 09 2025

A116161 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 8.

Original entry on oeis.org

3, 93, 145, 384, 505, 820, 993, 7680, 9993, 20665, 29748, 99993, 480340, 999993, 6694209, 9999993, 77854665, 99999993, 107541513, 109342209, 132231408, 174990384, 183673473, 213017748, 289940833, 326530608, 338353345

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116146, A116160, A116162, A116168, A116292.

A116277 n times n+8 gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

65, 738, 892, 755003, 789545, 3643102, 6356891, 9084164, 405460897734702722, 416974886220714236, 583025113779285757, 594539102265297271, 920312382883217574, 931826371369229088, 4330823773309769374

Offset: 1

Giovanni Resta, Feb 06 2006

			9084164 * 9084172 = 8252210//8252208, where // denotes
concatenation.

Cf. A116146, A116270, A116276, A116278, A116292.

A116293 Numbers k such that k * (k+9) is the concatenation of a number m with itself.

Original entry on oeis.org

2, 92, 420, 572, 693, 728, 992, 9855, 9992, 36355, 63637, 99992, 970298, 999992, 4545455, 5454537, 9999992, 88235295, 99999992, 351069983, 403018035, 493927126, 506072866, 596981957, 648930009, 736842097, 739839100, 766233767, 769230761, 827751188, 857142858, 860139852, 879699240, 909090910

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 09 2025: (Start)
Numbers k such that k * (k + 9) = (10^d + 1) * m for some d and m where m has d digits.
Contains 10^d-8 for all d >= 1. (End)

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

Cf. A116162, A116284, A116292, A116300.

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,9)),d=1..10)

Name edited and more terms from Robert Israel, Apr 09 2025

A116299 n times n+8 gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

40, 53, 40354307, 59645686, 39704957106129738595969799927610, 44505281604832422780051712184759, 45053875613995255103944518907119, 54946124386004744896055481092874, 55494718395167577219948287815234

Offset: 1

Giovanni Resta, Feb 06 2006

			59645686 * 59645694 = 35576083//35576084, where // denotes
concatenation.

Cf. A116168, A116292, A116298, A116300, A116305.

cp's OEIS Frontend

A116291 Numbers k such that k * (k + 7) is the concatenation of a number m with itself.

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A116161 Numbers k such that k concatenated with itself gives the product of two numbers which differ by 8.

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A116277 n times n+8 gives the concatenation of two numbers m and m-2.

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A116293 Numbers k such that k * (k+9) is the concatenation of a number m with itself.

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Author

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Links

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Programs

Maple

Extensions

A116299 n times n+8 gives the concatenation of two numbers m and m+1.

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Author

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