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.

A000092 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

Original entry on oeis.org

1, 2, 5, 6, 14, 21, 29, 30, 54, 90, 134, 155, 174, 230, 234, 251, 270, 342, 374, 461, 494, 550, 666, 750, 810, 990, 1890, 2070, 2486, 2757, 2966, 3150, 3566, 3630, 4554, 4829, 5670, 5750, 8154, 8382, 8774, 8910, 10350, 10710, 15734, 15750, 16302, 17550
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Indices n for which A210641(n) = A117609(n) - A210639(n) yields record values (in absolute value). - M. F. Hasler, Mar 26 2012

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000323, A000036, A000099, A000413, A000223.

Programs

  • Mathematica
    P[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi*n^(3/2)]; record = 0; A000092 = Reap[For[n=1, n <= 2*10^4, n++, If[(p = Abs[P[n]]) > record, record = p; Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2016, after M. F. Hasler *)
  • PARI
    m=0; for(n=1,1e4, if(m+0A210641(n)),m),print1(n",")))  /* Start with n=0 to print the initial 0. */ \\ M. F. Hasler, Mar 26 2012

Extensions

Revised Jun 28 2005

A000036 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives closest integer to P(A000099(n)).

Original entry on oeis.org

2, 3, 5, 6, 6, -6, 7, 8, 10, 13, 13, 13, 14, -17, 17, 17, 18, -19, 20, -22, 23, 27, -29, -29, 29, -31, -32, -35, 36, -37, -40, -43, -46, -48, -50, -53, -55, -57, -60, -60, -61, -63, -66, -66, -68, -71, -74, -77, -79, -82, -85, -88, -89, -92, -95, -96, -97, -97, -100
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000092, A000099, A000223, A000323, A000413.

Programs

  • Mathematica
    nmax = 6*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]] + 1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000036 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[pn = P[n]]; If[p > record, record = p; k++; Sow[pn // Round]; Print["a(", k, ") = ", pn // Round]]]][[2, 1]] (* Jean-François Alcover, Feb 03 2016 *)

Formula

a(n) = round(P(A000099(n))), where P(n) = A057655(n)-pi*n. - David W. Wilson, May 15 2008

Extensions

Revised by N. J. A. Sloane, Jun 26 2005
More terms from David W. Wilson, May 15 2008

A000223 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).

Original entry on oeis.org

3, 7, 10, 19, 32, 34, 37, 51, 81, 119, 122, 134, 157, 160, 161, 174, 221, 252, 254, 294, 305, 309, 364, 371, 405, 580, 682, 734, 756, 763, 776, 959, 1028, 1105, 1120, 1170, 1205, 1550, 1570, 1576, 1851, 1930, 2028, 2404, 2411, 2565, 2675, 2895, 2905, 2940, 3133, 3211, 3240, 3428
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Record values of (absolute values of) A210641 = A117609-A210639. It appears that the records occur always at positive elements of that sequence. (One could add an initial a(0)=1.) - M. F. Hasler, Mar 26 2012

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000323, A000036, A000092, A000413, A000099.

Programs

  • Mathematica
    nmax = 3*10^4; P[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi* n^(3/2)]; record = 0; A000223 = Reap[For[n = 1, n <= nmax, n++, If[(p = Abs[pn = P[n]]) > record, record = p; Print[pn]; Sow[pn]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2016 *)
  • PARI
    m=0;for(n=0,1e4, mA210641(n)) & print1(m=A210641(n)",")) /* This would print a negative value in case the record in absolute value occured for A117609(n)<A210639(n), which does not happen for n<10^4. */ \\ M. F. Hasler, Mar 26 2012

Formula

a(n) = |A210641(A000092(n))|. - M. F. Hasler, Mar 26 2012

Extensions

Revised Jun 28 2005

A000099 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

Original entry on oeis.org

1, 2, 5, 10, 20, 24, 26, 41, 53, 130, 149, 205, 234, 287, 340, 410, 425, 480, 586, 840, 850, 986, 1680, 1843, 2260, 2591, 3023, 3024, 3400, 3959, 3960, 5182, 5183, 7920, 9796, 11233, 14883, 15119, 15120, 19593, 21600, 21603, 21604, 22177, 28559, 28560
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000323, A000036, A000092, A000413, A000223.

Programs

  • Mathematica
    nmax = 3*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]]+1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000099 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[P[n]]; If[p > record, record = p; k++; Sow[n]; Print["a(", k, ") = ", n];]]][[2, 1]] (* Jean-François Alcover, Feb 03 2016 *)

Extensions

Entry revised by N. J. A. Sloane, Jun 26 2005

A000323 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives A(A000099(n)).

Original entry on oeis.org

5, 9, 21, 37, 69, 69, 89, 137, 177, 421, 481, 657, 749, 885, 1085, 1305, 1353, 1489, 1861, 2617, 2693, 3125, 5249, 5761, 7129, 8109, 9465, 9465, 10717, 12401, 12401, 16237, 16237, 24833, 30725, 35237, 46701, 47441, 47441, 61493, 67797, 67805, 67805
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000099, A000036, A000092, A000413, A000223.

Programs

  • Mathematica
    nmax = 3*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]] + 1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000099 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[P[n]]; If[p > record, record = p; k++; Sow[an = A[n]]; Print["a(", k, ") = ", an];]]][[2, 1]] (* Jean-François Alcover, Feb 07 2016*)

Extensions

Entry revised Jun 28 2005
Showing 1-5 of 5 results.

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

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