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 1-5 of 5 results.

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

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

Original entry on oeis.org

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

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

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)

Links

Crossrefs

Cf. A116154, A116272, A116286, A116294, A161356.

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

Formula

A161356(n) = a(n)*(a(n)+1). - Michael S. Branicky, Jul 11 2025

Extensions

More terms from Robert Israel, Apr 08 2025

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

A116295 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

8873, 9010, 83352841, 99000100, 329767122287, 670232877712, 738226276372, 933006600340, 999000001000, 3779410975143114, 3872816717528066, 4250291784692549, 4278630943941866, 4372036686326818, 4749511753491301
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Jun 06 2018: (Start)
Numbers k such that 10^m+1 | (k+1)^2-2 where (k+1)^2 has 2*m digits.
Includes 10^i - 10^(3*i) + 10^(4*i) for all i >= 1. (End)

Examples

			99000100 * 99000102 = 98010199//98010200, where // denotes concatenation.

Links

Crossrefs

Cf. A115426, A116286, A116294, A116296, A116308.

Programs

  • Maple
    Res:= NULL:
for d from 1 to 40 do
  Res:= Res, op(sort(select(t -> t^2 >= 10^(2*d-1),map(t -> rhs(op(t))-1,[msolve(x^2=2, 10^d+1)]))))
od:
Res; # Robert Israel, Jun 06 2018

A116279 Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

36363636363, 45454545454, 54545454545, 63636363636, 72727272727, 81818181818, 90909090909, 428571428571428571428, 571428571428571428571, 714285714285714285714, 857142857142857142857
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			36363636363 * 36363636365 = 13223140496//13223140495, where // denotes concatenation.

Crossrefs

Cf. A102567, A116273, A116280, A116286.
Showing 1-5 of 5 results.

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

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