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 20 results. Next

A295489 Numbers that have exactly six representations as a sum of six nonnegative squares.

Original entry on oeis.org

30, 33, 34, 35, 39, 40
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295670 Numbers that have exactly one representation as a sum of six positive squares.

Original entry on oeis.org

6, 9, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 31, 32, 34, 35, 37, 40, 43
Offset: 1

Views

Author

Robert Price, Nov 25 2017

Keywords

Comments

It appears that this sequence is finite and complete. See the von Eitzen link for a proof for the 5 positive squares case.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A025430, A243148, A294524.

Programs

  • Mathematica
    m = 6;
r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, Positive] && n == Total[xx^2], xx, Integers];
For[n = 0, n < 50, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* Jean-François Alcover, Feb 25 2019 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 6], {n, 0, 100}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

Formula

A243148(a(n),6) = 1. - Alois P. Heinz, Feb 25 2019

A295485 Numbers that have exactly two representations as a sum of six nonnegative squares.

Original entry on oeis.org

4, 5, 6, 8, 10, 11, 15
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295486 Numbers that have exactly three representations as a sum of six nonnegative squares.

Original entry on oeis.org

9, 12, 13, 14, 16, 19, 23
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295487 Numbers that have exactly four representations as a sum of six nonnegative squares.

Original entry on oeis.org

17, 18, 22, 24, 31
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295488 Numbers that have exactly five representations as a sum of six nonnegative squares.

Original entry on oeis.org

20, 21, 25, 26, 27, 28, 32
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295490 Numbers that have exactly seven representations as a sum of six nonnegative squares.

Original entry on oeis.org

29, 37, 42, 43, 47, 48
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295491 Numbers that have exactly eight representations as a sum of six nonnegative squares.

Original entry on oeis.org

38, 46, 55
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295492 Numbers that have exactly nine representations as a sum of six nonnegative squares.

Original entry on oeis.org

36, 41, 44, 49, 51, 64
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 6501, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least 10. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 6501 is at least ten. Then an inspection of n <= 6501 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.

A295493 Numbers that have exactly ten representations as a sum of six nonnegative squares.

Original entry on oeis.org

45, 50, 56, 58
Offset: 1

Views

Author

Robert Price, Nov 22 2017

Keywords

Comments

This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 7845, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least 11. Since this sequence relaxes the restriction of zero squares and allows one more square, the number of representations for n > 7845 is at least 11. Then an inspection of n <= 7845 completes the proof.

References

  • E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

Crossrefs

Cf. A000177, A294524, A295150.
Showing 1-10 of 20 results. Next

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