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.

A082210 Square root of the squares arising in A082209.

Original entry on oeis.org

1, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248, 71, 4, 8, 7, 31, 248
Offset: 1

Views

Author

Amarnath Murthy, Apr 10 2003

Keywords

Comments

The squares are given in A090567. - M. F. Hasler, Nov 24 2015

Examples

			a(3) = 31 = 961^(1/2) where A082209(3) = 9 and A082209(4) = 61.

Links

Crossrefs

Cf. A082209.

Programs

  • Mathematica
    Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 1},{4, 8, 7, 31, 248, 71},77]] (* Ray Chandler, Aug 26 2015 *)
  • PARI
    A082210(n)=n<2||return([248,71,4,8,7,31][n%6+1]) \\ M. F. Hasler, Nov 24 2015

Formula

a(n) = {A082209(n) concatenated with A082209(n+1)}^(1/2).
Periodic with period 6 from n=2 onward. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
a(n) = sqrt(A090567(n)) with A090567(n) = A082209(n) concat A243091(A082209(n)). - M. F. Hasler, Nov 24 2015

Extensions

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
Definition corrected and minor edits by M. F. Hasler, Nov 24 2015

A090567 Squares arising in A082209.

Original entry on oeis.org

16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49, 961, 61504, 5041, 16, 64, 49
Offset: 1

Views

Author

Amarnath Murthy, Dec 11 2003

Keywords

Programs

Formula

a(n) = A082210(n+1)^2. - David Wasserman, Jan 05 2006

Extensions

Corrected and extended by David Wasserman, Jan 05 2006

A090566 a(1) = 1; thereafter a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.

Original entry on oeis.org

1, 6, 25, 281, 961, 6201, 59409, 187600, 730641, 4429444, 28600025, 85336064, 468650384, 4590568025, 23901253604, 36922256164, 228378872384, 519390415729, 3999576229761, 22053449580964, 52752598923921, 67153745961316, 346596997521321, 2205389504844676, 32117901134901281
Offset: 1

Views

Author

Amarnath Murthy, Dec 11 2003

Keywords

Comments

From David W. Wilson, Nov 22 2015: (Start)
I used the following algorithm to extend the sequence:
x = a(n);
d = number of digits in x;
p = (10^d + 1)*x; #concat(x, x)
q = (floor(sqrt(p)) + 1)^2; #smallest square > p
if (q < (10^d)(x + 1))
a(n+1) = q mod (10^d); #last d digits of q
else
p = (10^d)*(10x + 1); #concat(x, 10^d)
q = (floor(sqrt(p - 1)) + 1)^2; #smallest square >= p
a(n+1) = q mod (10^(d + 1)); #last d+1 digits of q.
(End)

Links

Crossrefs

See A082209 for another version. Cf. A243091.

Programs

  • Maple
    A[1]:= 1:
for n from 2 to 100 do
  x:= A[n-1];
  d:= ilog10(x)+1;
  for dp from d while not assigned(A[n]) do
    if dp = d then
      ymin:= x+1
    else
      ymin:= 10^(dp-1)
    fi;
    zmin:= 10^dp*x + ymin;
    r:= isqrt(zmin);
    if r^2 < zmin then z:= (r+1)^2
    else z:= r^2
    fi;
    if z <= 10^dp*x + 10^dp - 1 then
        A[n]:= z - 10^dp*x;
    fi
  od
od:
seq(A[i],i=1..100); # Robert Israel, Nov 22 2015
  • Mathematica
    a[1] = 1; a[n_] := Block[{x = a[n - 1], d = 1 + Floor@ Log10@ a[n - 1]}, q = (Floor@ Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@ Sqrt[(10^d)(10x + 1) -1] + 1)^2, 10^(d + 1)]]]; Array[a, 25] (* after the algorithm of David W. Wilson, Robert G. Wilson v, Nov 22 2015 *)
  • PARI
    A090566(n,show=0,a=1)={for(i=2,n,show&&print1(a","); a=A243091(a));a} \\ Use 2nd optional arg to print out intermediate values, 3rd optional arg to use another starting value. - M. F. Hasler, Nov 22 2015, revised version based on A243091: Nov 24 2015

Formula

a(n+1) = A243091(a(n)). - M. F. Hasler, Nov 24 2015

Extensions

Corrected and extended by David W. Wilson, Nov 20 2015

A264770 a(1) = 1, a(n) = smallest positive number not yet in the sequence such that the concatenation of a(n-1) and a(n) is a square.

Original entry on oeis.org

1, 6, 4, 9, 61, 504, 100, 489, 2944, 656, 3844, 34449, 85636, 516, 961, 6201, 5625, 43524, 36729, 7225, 344, 569, 2996, 361, 201, 64, 1601, 6004, 7001, 316, 84, 681, 21, 16, 81, 225, 625, 5001, 3184, 3449, 2129, 8225, 424, 36, 481, 636, 804, 609, 1024, 144
Offset: 1

Views

Author

Robert Israel, Nov 24 2015, following a suggestion from N. J. A. Sloane

Keywords

Comments

For any x > 0, if d is large enough there are squares between 10^d*x + 10^(d-1) and 10^d*x + 10^d - 1. Thus the sequence is infinite.
a(3) = 4 is the minimum value of a(n) for n > 1. - Altug Alkan, Nov 24 2015 (This is because no square can end in 2 or 3, so 2 and 3 can never appear in the sequence. - N. J. A. Sloane, Nov 24 2015)

Examples

			For n = 6, a(n-1) = 61.  There are no squares of the form 61x or 61xy with x>=1.  The least square of the form 61xyz with x >= 1 is 61504, and 504 has not appeared previously so a(6) = 504.

Links

Crossrefs

Cf. A082209, A090566.

Programs

  • Maple
    S:= {1};
A[1]:= 1;
for n from 2 to 100 do
   found:= false;
   x:= A[n-1];
   for d from 1 while not found do
      a:= ceil(sqrt(10^d*x +10^(d-1)));
      b:= floor(sqrt(10^d*x + 10^d - 1));
      Q:= map(t -> t^2 - 10^d*x, {$a..b}) minus S;
      if nops(Q) >= 1 then
         A[n]:= min(Q);
         S:= S union {A[n]};
         found:= true;
      fi
   od
od:
seq(A[n],n=1..100);
  • Mathematica
    (*to get B numbers of the sequence*) A={1};i=1;While[iEmmanuel Vantieghem, Nov 24 2015 *)
  • PARI
    A264770(n,show=0,a=1,u=[])={for(n=2, n, u=setunion(u,[a]); show&&print1(a", "); my(k=3); until(!setsearch(u, k++) && issquare(eval(Str(a,k))),);a=k); a} \\ Use optional 2nd, 3rd or 4th argument to print intermediate terms, use another starting value, or exclude some numbers. - M. F. Hasler, Nov 24 2015

A264775 Square root of concatenation(A090566(n), A090566(n+1)).

Original entry on oeis.org

4, 25, 159, 531, 3101, 24903, 243740, 433129, 2703038, 21046245, 53478992, 292123372, 2164833445, 21425610902, 48888908358, 192151648872, 477890021223, 2279013856319, 19998940546342, 46961100477911, 72630984382646, 259140398165389, 1861711571434526, 14850553878036591, 56672657547446354, 303286443000496342, 2448230500494961492, 4821616822771036069, 24316253937847771953
Offset: 1

Views

Author

M. F. Hasler, Nov 24 2015

Keywords

Comments

Sequence A090566 lists the smallest integer larger than the preceding term such that their concatenation is a square. This sequence lists the square root of these squares.
The initial term a(0)=1 corresponds to the natural extension A090566(0):=0 (or an "empty" A090566(0)).
Sequence A082210 is the analog of this corresponding to the variant A082209 of A090566.

Links

Crossrefs

Cf. A090566; A082210, A082209.
Showing 1-5 of 5 results.

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

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