A000177 -id:A000177 - cp's OEIS frontend

A295697 Numbers that have exactly seven representations as a sum of six positive squares.

60, 65, 68, 69, 77, 90, 112

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.

A295698 Numbers that have exactly eight representations as a sum of six positive squares.

Original entry on oeis.org

87, 94, 96, 97, 98, 103, 109

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.

A295699 Numbers that have exactly nine representations as a sum of six positive squares.

Original entry on oeis.org

78, 99, 115

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.

A295700 Numbers that have exactly ten representations as a sum of six positive squares.

Original entry on oeis.org

81, 86, 93, 95, 100, 104, 107, 114, 130, 133

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.

A295484 Numbers that have exactly one representation as a sum of six nonnegative squares.

Original entry on oeis.org

0, 1, 2, 3, 7

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.

A295701 Smallest number with exactly n representations as a sum of six positive squares.

Original entry on oeis.org

0, 2006, 30611

Offset: 0

Robert Price, Nov 25 2017

nonn

It appears that this sequence is finite and complete.

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

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.

cp's OEIS Frontend

A295697 Numbers that have exactly seven representations as a sum of six positive squares.

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A295698 Numbers that have exactly eight representations as a sum of six positive squares.

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A295699 Numbers that have exactly nine representations as a sum of six positive squares.

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A295700 Numbers that have exactly ten representations as a sum of six positive squares.

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A295484 Numbers that have exactly one representation as a sum of six nonnegative squares.

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A295701 Smallest number with exactly n representations as a sum of six positive squares.

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