A000177 -id:A000177 - cp's OEIS frontend

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

20, 21, 25, 26, 27, 28, 32

Offset: 1

Robert Price, Nov 22 2017

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.

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

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

Robert Price, Nov 22 2017

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.

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

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

Robert Price, Nov 22 2017

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.

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

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

Robert Price, Nov 22 2017

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.

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

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

Robert Price, Nov 22 2017

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.

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A294524, A295150.

A295692 Numbers that have exactly two representations as a sum of six positive squares.

Original entry on oeis.org

21, 24, 29, 42, 58, 64

Offset: 1

Robert Price, Nov 25 2017

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

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A025430, A294524.

A295693 Numbers that have exactly three representations as a sum of six positive squares.

Original entry on oeis.org

30, 33, 38, 39, 46, 47, 48, 49, 50, 51, 52, 55, 59, 61, 67

Offset: 1

Robert Price, Nov 25 2017

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

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A025430, A294524.

A295694 Numbers that have exactly four representations as a sum of six positive squares.

Original entry on oeis.org

36, 41, 44, 45, 53, 56, 82

Offset: 1

Robert Price, Nov 25 2017

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

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A025430, A294524.

A295695 Numbers that have exactly five representations as a sum of six positive squares.

Original entry on oeis.org

63, 66, 70, 73, 74, 79, 85, 91

Offset: 1

Robert Price, Nov 25 2017

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

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A025430, A294524.

A295696 Numbers that have exactly six representations as a sum of six positive squares.

Original entry on oeis.org

54, 57, 62, 71, 72, 75, 76, 80, 83, 88, 106

Offset: 1

Robert Price, Nov 25 2017

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

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

H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000177, A025430, A294524.

cp's OEIS Frontend

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

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

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

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

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

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A295692 Numbers that have exactly two representations as a sum of six positive squares.

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A295693 Numbers that have exactly three representations as a sum of six positive squares.

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A295694 Numbers that have exactly four representations as a sum of six positive squares.

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A295695 Numbers that have exactly five representations as a sum of six positive squares.

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A295696 Numbers that have exactly six representations as a sum of six positive squares.

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