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.

A116168 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 8.

Original entry on oeis.org

19, 32, 16284704, 35576083, 15764836187996024260119639732979, 19807200907254352332962649366152, 20298517078413563250826300137112, 30190765850423053042937262322867, 30796637697589506772859224996627
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Also numbers k such that k concatenated with k+8 gives the product of two numbers which differ by 6.

Examples

			35576083//35576084 = 59645686 * 59645694, where // denotes concatenation.
35576083//35576091 = 59645687 * 59645693.

Crossrefs

Cf. A116161, A116167, A116169, A116174, A116299.
Cf. A116208, A115430, A116215, A116345.

Extensions

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

A116292 Numbers k such that k * (k + 8) is the concatenation of a number m with itself.

Original entry on oeis.org

3, 93, 377, 616, 707, 902, 993, 8760, 9993, 45455, 54538, 99993, 693062, 999993, 8181811, 9999993, 88235287, 99999993, 327935223, 330669331, 363636364, 418318516, 428571429, 461538454, 538461539, 571428564, 581681477, 636363629, 669330662, 672064770, 691571587, 756506652, 781954880, 789473685
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

From Robert Israel, Apr 09 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-7 for all d >= 1. (End)

Links

Crossrefs

Cf. A116161, A116277, A116291, A116293, A116299.

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,8)),d=1..10); # Robert Israel, Apr 09 2025
  • Mathematica
    ccnQ[n_]:=With[{ccc=With[{c=n(n+8)},TakeDrop[IntegerDigits[c],IntegerLength[c]/2]]},ccc[[1]]==ccc[[2]]];  Select[Range[10^6],ccnQ]//Quiet (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Jul 05 2025 *)

Extensions

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

A116298 n times n+7 gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

6, 699, 776, 790, 867, 42337, 57657, 96883, 44121666, 55878328, 85298137, 36680703009575609347721358493, 63319296990424390652278641501, 69346342454876071597336150481, 81501115172242572470460459683
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			85298137 * 85298144 = 72757727//72757728, where // denotes
concatenation.

Crossrefs

Cf. A116167, A116291, A116297, A116299, A116311.

A116300 n times n+9 gives the concatenation of two numbers m and m+1.

Original entry on oeis.org

26, 66, 3416102887775247376839416334668635, 3756559953325598880263233435801764, 4313503800489302411917772257282208
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			66 * 75 = 49//50, where // denotes concatenation.

Links

Crossrefs

Cf. A116169, A116293, A116299, A116306.

Programs

  • Maple
    F:= proc(d) local t, g, Cands:
  t:= 10^d+1;
  if NumberTheory:-QuadraticResidue(85,t) <> 1 then return NULL fi;
  Cands:= map(s -> rhs(op(s)), [msolve(x^2 + 9*x - 1, t)]);
  g:= proc(r) local v; v:= r^2 + 9*r - 1; v >= t*(t-11)/10 and v < t*(t-2) end proc;
  op(sort(select(g, Cands)));
end proc:
map
map(F, [$1..82]); # Robert Israel, Aug 25 2023
  • Python
    from itertools import count, islice
from sympy import sqrt_mod_iter
def A116300_gen(): # generator of terms
    for l in count(1):
        m = 10**l+1
        k, r, dlist = m*(m-11)/10, m*(m-2), []
        for a in sqrt_mod_iter(85,m):
            d = ((a if a&1 else a+m)>>1)-4
            if kA116300_list = list(islice(A116300_gen(),14)) # Chai Wah Wu, May 07 2024

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

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

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

Crossrefs

Cf. A116174, A116299, A116304, A116306, A116312.
Showing 1-5 of 5 results.

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

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