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 12 results. Next

A084415 Erroneous version of A076600.

Original entry on oeis.org

4, 12, 8, 24, 15, 12, 24, 60, 16, 84, 48, 20, 30, 144, 24, 180, 21, 28, 120, 264, 32, 60
Offset: 3

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Author

Keywords

A076671 Smallest a(n) > a(n-1) such that a(n)^2+a(n-1)^2 is a perfect square, with a(1)=5.

Original entry on oeis.org

5, 12, 16, 30, 40, 42, 56, 90, 120, 126, 168, 224, 360, 378, 504, 550, 1320, 1386, 1848, 1989, 2652, 2961, 3948, 5264, 8052, 9711, 12948, 17264, 24852, 31311, 41748, 53289, 71052, 94736, 130548, 145061, 146280, 153594, 163392, 170280, 173290
Offset: 1

Views

Author

Zak Seidov, Oct 25 2002

Keywords

Comments

The sequence is infinite.
If we require the terms to be distinct, but not necessarily increasing, then the sequence "paints itself into a corner" and can't be continued: 5, 12, 9, 40, 30, 16, 63, 60, 11. - Ivan Neretin, Dec 15 2016

Links

Crossrefs

Cf. A076600.

Programs

  • Mathematica
    Nest[Append[#, k = #[[-1]]; d = Divisors[k^2]; Min@Select[(Reverse@d - d)/2, IntegerQ@# && # > k &]] &, {5}, 40] (* Ivan Neretin, Dec 15 2016 *)

A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

Original entry on oeis.org

0, 16, 9, 0, 20, 12, 441, 64, 16, 90, 1089, 36, 4212, 98, 225, 0, 272, 144, 549081, 25, 567, 2156, 13225, 48, 144, 650, 81, 98, 142100, 150, 71622369, 256, 363, 578, 1225, 64, 1332, 684, 468, 360, 41984, 252, 521345889, 198, 180, 559682, 108241, 144, 63, 400, 127449, 117, 1755572, 108, 2420, 392, 4275, 568458
Offset: 1

Views

Author

Felix Huber, Feb 07 2025

Keywords

Comments

a(1) = a(4) = a(16) = 0. Proof: See Huber link.
k > n exists for n > 16.

Examples

			a(3) = 9 because sqrt(1/3 + 1/4) = sqrt(7/12) is irrational, sqrt(1/3 + 1/5) = sqrt(8/15) is irrational, sqrt(1/3 + 1/6) = sqrt(1/2) is irrational, sqrt(1/3 + 1/7) = sqrt(10/21) is irrational, sqrt(1/3 + 1/8) = sqrt(11/24) is irrational, but sqrt(1/3 + 1/9) = sqrt(4/9) = 2/3 is rational.

Links

Crossrefs

Cf. A002350, A024352, A076600, A378501, A379816.
Cf. A357372, A257522.

Programs

  • Maple
    A379815:=proc(n)
    local k;
    if n=1 or n=4 or n=16 then
        return 0
    else
        for k from n+1 do
            if type(sqrt(1/n+1/k),rational) then
                return k
            fi
        od
    fi;
end proc;
seq(A379815(n),n=1..58);
  • PARI
    a(n) = if ((n==1) || (n==4) || (n==16), return(0)); my(k=n+1); while (!issquare(1/n + 1/k), k++); k; \\ Michel Marcus, Feb 08 2025

Formula

a(n) <= n*A002350(n)^2 - n if n is not a square; a(m^2) <= A076600(m)^2. - Jinyuan Wang, Feb 11 2025

A077059 Numbers j such that j^2 + k^2 is a cube for some k > j.

Original entry on oeis.org

0, 2, 5, 7, 9, 10, 16, 17, 18, 26, 30, 35, 36, 37, 38, 40, 44, 47, 50, 51, 54, 56, 58, 60, 65, 72, 73, 74, 75, 80, 82, 88, 95, 97, 101, 102, 106, 107, 115, 122, 128, 130, 135, 136, 140, 142, 143, 144, 145, 146, 150, 154, 164, 170, 174, 182, 187, 189, 190
Offset: 1

Views

Author

Zak Seidov, Oct 23 2002, Jan 02 2008

Keywords

Comments

The values of minimal k in A077060 and A077061. The values of corresponding c in A077062.

Examples

			2 is in the sequence because 2^2 + 11^2 = 5^3;
7 is in the sequence because 7^2 + 524^2 = 65^3;
1 is not in the sequence because 1^2 + k^2 is not a cube for any k > 1;
3 is not in the sequence because 3^2 + k^2 is not a cube for any k > 3.
174 is in the sequence because 174^2 + 406^2 = 58^3. - _Sean A. Irvine_, May 02 2025

Crossrefs

Cf. A076600, A077060, A077061, A077062.

Formula

j is in this sequence if for some integer k > j, j^2 + k^2 = c^3.

Extensions

Corrected and extended by Max Alekseyev, Dec 14 2011
Missing 174 inserted by Sean A. Irvine, May 02 2025

A077060 Smallest k>A077059(n) such that A077059(n)^2+k^2 is a cube (=A077061(n)^3).

Original entry on oeis.org

8, 11, 10, 524, 46, 30, 88, 68, 26, 39, 2935, 120, 2681, 222, 10582, 80, 117, 52, 350, 782, 297, 4192, 145, 1745, 142, 368, 77964, 110, 100, 240, 738, 835, 1104490, 1405096, 1010, 170, 371, 198, 236, 1342, 704, 56303, 270, 544, 505, 551122, 1962
Offset: 1

Views

Author

Zak Seidov, Oct 23 2002

Keywords

Examples

			k(0)=8 because A077059(0)^2+k(0)^2=0^2+8^2=4^3; k(0)=1 if 1 is a full cube.

Crossrefs

Cf. A076600, A077059-A077062.

Formula

n^2+k^2=c^3, k>n.

Extensions

Corrected and extended by Max Alekseyev, Dec 14 2011

A077061 a(n) = ( A077059(n)^2 + A077060(n)^2 )^(1/3).

Original entry on oeis.org

4, 5, 5, 65, 13, 10, 20, 17, 10, 13, 205, 25, 193, 37, 482, 20, 25, 17, 50, 85, 45, 260, 29, 145, 29, 52, 1825, 26, 25, 40, 82, 89, 10685, 12545, 101, 34, 53, 37, 41, 122, 80, 1469, 45, 68, 65, 6722, 157, 40, 125, 50, 250, 58, 41, 50, 58, 74, 389, 585
Offset: 1

Views

Author

Zak Seidov, Oct 23 2002

Keywords

Examples

			a(1)=4 because A077059(1)^2+A077060(1)^2=0^2+8^2=4^3.

Crossrefs

Cf. A076600, A077059, A077060, A077062.

Extensions

Corrected and extended by Max Alekseyev, Dec 14 2011
Missing a(55) inserted by Sean A. Irvine, May 02 2025

A077062 Minimal k > n such that n^2 + k^2 is a cube for k > n, or 0 if no such k exists.

Original entry on oeis.org

8, 0, 11, 0, 0, 10, 0, 524, 0, 46, 30, 0, 0, 0, 0, 0, 88, 68, 26, 0, 0, 0, 0, 0, 0, 0, 39, 0, 0, 0, 2935, 0, 0, 0, 0, 120, 2681, 222, 10582, 0, 80, 0, 0, 0, 117, 0, 0, 52, 0, 0, 350, 782, 0, 0, 297, 0, 4192, 0, 145, 0, 1745, 0, 0, 0, 0, 142, 0, 0, 0
Offset: 0

Views

Author

Zak Seidov, Oct 23 2002

Keywords

Examples

			a(0)=8 because 0^2 + 8^2 = 4^3;
a(2)=11 because 2^2 + 11^2 = 5^3.

Crossrefs

Cf. A076600, A077059, A077060, A077061.

Extensions

Corrected and extended by Max Alekseyev, Dec 14 2011
Missing a(58) inserted by Sean A. Irvine, May 02 2025

A076673 Smallest a(n)>a(n-1) such that a(n)^2+a(n-1)^2 is a perfect square, a(1)=7.

Original entry on oeis.org

7, 24, 32, 60, 63, 84, 112, 180, 189, 252, 275, 660, 693, 924, 1232, 1326, 1768, 1974, 2632, 4026, 5368, 6405, 8200, 8319, 11092, 11715, 15620, 16401, 19720, 20706, 20880, 20910, 24752, 24960, 25300, 26565, 29716, 29835, 33048, 35055, 41496
Offset: 1

Views

Author

Zak Seidov, Oct 25 2002

Keywords

Comments

The sequence is infinite.

Crossrefs

Cf. A076600.

Programs

  • Mathematica
    nxt[n_]:= Module[{k = n + 1}, While[!IntegerQ[Sqrt[n^2 + k^2]], k++]; k]; NestList[nxt,7,40] (* Harvey P. Dale, May 29 2015 *)

A076676 Smallest a(n)>a(n-1) such that a(n)^2+a(n-1)^2 is a perfect square, a(1)=11.

Original entry on oeis.org

11, 60, 63, 84, 112, 180, 189, 252, 275, 660, 693, 924, 1232, 1326, 1768, 1974, 2632, 4026, 5368, 6405, 8200, 8319, 11092, 11715, 15620, 16401, 19720, 20706, 20880, 20910, 24752, 24960, 25300, 26565, 29716, 29835, 33048, 35055, 41496, 42997
Offset: 1

Views

Author

Zak Seidov, Oct 25 2002

Keywords

Comments

The sequence is infinite.

Links

Crossrefs

Cf. A076600.

Programs

  • Maple
    A076600:= proc(n) local q;
  q:= max(select(t -> n^2/t - t > 2*n and (t - n^2/t)::even, numtheory:-divisors(n^2)));
  if q = -infinity then 0 else (n^2/q - q)/2 fi;
end proc:
A[1]:= 11;
for n from 2 to 100 do
  A[n]:= A076600(A[n-1]);
od:
seq(A[i],i=1..100); # Robert Israel, Mar 22 2018
  • Mathematica
    nmax = 100;
A076600[n_] := Module[{q},
  q = Max[Select[Divisors[n^2], n^2/# - # > 2n &&
  EvenQ[# - n^2/#]&]];
  If[q == -Infinity, 0, (n^2/q - q)/2]];
a[1] = 11;
For[n = 2, n <= nmax, n++, a[n] = A076600[a[n - 1]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, May 17 2023, after Robert Israel *)

A076672 Smallest a(n)>a(n-1) such that a(n)^2+a(n-1)^2 is a perfect square, a(1)=6.

Original entry on oeis.org

6, 8, 15, 20, 21, 28, 45, 60, 63, 84, 112, 180, 189, 252, 275, 660, 693, 924, 1232, 1326, 1768, 1974, 2632, 4026, 5368, 6405, 8200, 8319, 11092, 11715, 15620, 16401, 19720, 20706, 20880, 20910, 24752, 24960, 25300, 26565, 29716
Offset: 1

Views

Author

Zak Seidov, Oct 25 2002

Keywords

Comments

The sequence is infinite.

Crossrefs

Cf. A076600, A076671 (starting at 5).

Programs

  • Mathematica
    Clear[nxt]; nxt[n_]:=Module[{i=n+1},While[!IntegerQ[Sqrt[n^2+i^2]],i++]; i]; NestList[nxt,6,40] (* Harvey P. Dale, Dec 03 2010 *)
Showing 1-10 of 12 results. Next

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