cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A116099 Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 7.

Original entry on oeis.org

69, 59898667, 79493157, 13412927190959690154913903, 14163000698458955079906403, 38895475965785687555173929, 40165600438484442828161229, 74294440818366638194239027
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 5.
Also numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 3.
Also numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 1.

Examples

			79493157//79493154 = 89158933 * 89158938, where // denotes concatenation.
79493157//79493158 = 89158934 * 89158937.
79493157//79493160 = 89158935 * 89158936.
79493157//79493148 = 89158932 * 89158939.

Crossrefs

Cf. A116098, A116193, A116107, A116230, A116130, A116136, A116205, A116107, A116268.
Cf. A115426, A115428, A116171, A116296, A116170, A116177, A116307.

Extensions

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

A116136 Numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 4.

Original entry on oeis.org

9, 99, 183, 328, 528, 715, 999, 6099, 9999, 13224, 40495, 99999, 106755, 453288, 999999, 2066115, 2975208, 9999999, 22145328, 28027683, 99999999, 110213248, 110667555, 147928995, 178838403, 226123528, 275074575, 333052608, 378698224, 445332888, 446245635, 518348515
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with itself gives the product of two numbers which differ by 2.

Examples

			8315420899//8315420896 = 9118892968 * 9118892972, where // denotes concatenation.

Links

Crossrefs

Cf. A116129, A115431, A116099, A115426, A116267.
Cf. also A102567, A116154, A116130, A116286.

Programs

  • Python
    from itertools import count, islice
from sympy import sqrt_mod
def A116136_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(1,a,all_roots=True)):
            if a*(b+3) <= k**2-1 < a*(a+2):
                yield (k**2-1)//a
A116136_list = list(islice(A116136_gen(),40)) # Chai Wah Wu, Feb 19 2024

Extensions

Edited by N. J. A. Sloane, Apr 15 2007
a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116125 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 6.

Original entry on oeis.org

7, 97, 205, 300, 477, 732, 997, 1920, 3157, 9997, 52896, 99997, 120085, 427020, 999997, 8264460, 9999997, 88581312, 99999997, 112000885, 112917765, 143075580, 152863360, 193537077, 233788192, 266755221, 313680096
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

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

Examples

			6786111717//6786111712 = 8237785936 * 8237785942, where // denotes concatenation.

Crossrefs

Cf. A115426, A116124, A116126, A116131, A116256.
Cf. A116130, A116158, A115428, A116288.

Extensions

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

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

Original entry on oeis.org

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

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A116154, A115426, A116170, A116294.
Cf. A116143, A102567, A116112, A116130, A116280.

Programs

  • Mathematica
    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 *)

Extensions

Edited by N. J. A. Sloane, Apr 15 2007, Jun 27 2009
More terms from Giovanni Resta, Jul 08 2018

A116098 Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 6.

Original entry on oeis.org

11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 100000000001, 1000000000001
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Aug 13 2018: (Start)
Contained in, and apparently identical, to A116129.
Numbers k such that k*(10^d+1) is a square, where k-9 has d decimal digits.
(End)

Examples

			100000001//99999992 = 99999998 * 100000004, where // denotes
concatenation.

Links

Crossrefs

Cf. A116097, A116099, A116106, A054214, A116128, A116129, A116130, A116136, A116229, A116260.

Programs

  • Maple
    g:= proc(d) local r,c,a,b;
   r:= mul(t[1],t=select(s -> s[2]::odd, ifactors(10^d+1)[2]))
   c:= ceil((10^(d-1)+9)/r);
   a:= isqrt(c);
   if a^2 < c then a:= a+1 fi;
   c:= floor((10^d+8)/r);
   b:= isqrt(c);
   if b^2 > c then b:= b-1 fi;
   seq(r*y^2, y = a..b)
end proc:
seq(g(d),d=1..60); # Robert Israel, Aug 13 2018

A116287 Numbers k such that k*(k+3) gives the concatenation of a number m with itself.

Original entry on oeis.org

8, 98, 767, 858, 910, 998, 3285, 6713, 9998, 45452, 54546, 99998, 990100, 999998, 8181819, 9999998, 70588233, 99999998, 343130554, 362637363, 363636361, 420053631, 421052632, 497975709, 502024289, 578947366, 579946367, 636363637, 637362635, 656869444, 706766918, 713286714, 714285712, 783689995
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

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

Links

Crossrefs

Cf. A116130, A116280, A116286, A116288, A116296.

Programs

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

Extensions

More terms from Robert Israel, Apr 08 2025

A116129 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 4.

Original entry on oeis.org

11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 100000000001, 1000000000001
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Aug 13 2018: (Start)
Contains, and appears to be identical to, A116098.
Numbers k such that (10^d+1)*k is a square, where k-4 has d digits. (End)

Examples

			100000001//99999997 = 99999999 * 100000003, where // denotes concatenation.

Crossrefs

Cf. A116098, A054214, A116128, A116130, A116136, A116260.

Programs

  • Maple
    g:= proc(d) local r,c,a,b;
   r:= mul(t[1],t=select(s -> s[2]::odd, ifactors(10^d+1)[2]));
   c:= ceil((10^(d-1)+4)/r);
   a:= isqrt(c);
   if a^2 < c then a:= a+1 fi;
   c:= floor((10^d+3)/r);
   b:= isqrt(c);
   if b^2 > c then b:= b-1 fi;
   seq(r*y^2, y = a..b)
end proc:
map(g, [$1..60]); # Robert Israel, Aug 13 2018

A116131 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 6.

Original entry on oeis.org

20, 31, 14564, 38239, 69919, 120395, 426436, 902596, 7478020, 9090220, 6671332084, 8114264059, 8482227259, 9900250996, 2244338786836, 2490577152964, 2509440638591, 2769448208395, 7012067592220
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			9900250996//9900250992 = 9950000498 * 9950000504, where // denotes concatenation.

Crossrefs

Cf. A116125, A116130, A116132, A115429, A116262.

A116261 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

7, 97, 766, 857, 909, 997, 3284, 6712, 9997, 45451, 54545, 99997, 990099, 999997, 8181818, 9999997, 70588232, 99999997, 343130553, 362637362, 363636360, 420053630, 421052631, 497975708, 502024288, 578947365, 579946366
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Crossrefs

Cf. A028557, A116130, A116255, A116260, A116262, A116268.

A116124 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 5.

Original entry on oeis.org

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

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			9225507211//9225507206 = 9604950394 * 9604950399, where // denotes concatenation.

Crossrefs

Cf. A116118, A054214, A116125, A116130, A116255.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.