A287166 -id:A287166 - cp's OEIS frontend

A287165 Smallest number with exactly n representations as a sum of 6 nonzero squares or 0 if no such number exists.

6, 21, 30, 36, 63, 54, 60, 87, 78, 81, 84, 111, 102, 117, 108, 116, 126, 129, 134, 137, 132, 150, 172, 165, 161, 156, 177, 164, 195, 191, 182, 213, 180, 188, 198, 0, 204, 206, 215, 222, 243, 212, 251, 262, 233, 230

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 6 because 6 is the smallest number with exactly 1 representation as a sum of 6 nonzero squares: 6 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 21 because 21 is the smallest number with exactly 2 representations as a sum of 6 nonzero squares: 21 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2, etc.

Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A016032, A025414, A025416, A025430, A080654, A287166, A287167.

A025430(a(n)) = n for a(n) > 0.

A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

Original entry on oeis.org

8, 23, 35, 32, 46, 58, 72, 56, 62, 70, 71, 79, 80, 83, 88, 89, 91, 86, 103, 94, 109, 104, 107, 112, 113, 110, 122, 119, 126, 121, 118, 144, 0, 128, 131, 136, 137, 153, 143, 139, 149, 134, 0, 0, 142, 152, 164, 154

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 8 because 8 is the smallest number with exactly 1 representation as a sum of 8 nonzero squares: 8 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 23 because 23 is the smallest number with exactly 2 representations as a sum of 8 nonzero squares: 23 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2, etc.

Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A016032, A025414, A025416, A025432, A080654, A287165, A287166.

A025432(a(n)) = n for a(n) > 0.

A295797 Numbers that have exactly one representation as a sum of seven positive squares.

Original entry on oeis.org

7, 10, 13, 15, 16, 18, 19, 21, 23, 24, 26, 27, 29, 32, 35, 36, 41, 44

Offset: 1

Robert Price, Nov 27 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. A025431, A243148, A287166, A295670.

m = 7;
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, 7], {n, 0, 100}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

A243148(a(n),7) = 1. - Alois P. Heinz, Feb 25 2019

A295799 Numbers that have exactly two representations as a sum of seven positive squares.

Original entry on oeis.org

22, 25, 28, 30, 33, 38

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295692.

A295800 Numbers that have exactly three representations as a sum of seven positive squares.

Original entry on oeis.org

31, 34, 39, 43, 47, 51, 56, 59, 68

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295693.

A295801 Numbers that have exactly four representations as a sum of seven positive squares.

Original entry on oeis.org

37, 40, 42, 46, 48, 49, 50, 52, 53, 62, 65

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295694.

A295802 Numbers that have exactly five representations as a sum of seven positive squares.

Original entry on oeis.org

45, 54, 57, 60

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295695.

A295803 Numbers that have exactly six representations as a sum of seven positive squares.

Original entry on oeis.org

67, 71, 83

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295696.

A295804 Numbers that have exactly seven representations as a sum of seven positive squares.

Original entry on oeis.org

55, 58, 63, 64, 74, 75, 80, 89

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295697.

A295805 Numbers that have exactly eight representations as a sum of seven positive squares.

Original entry on oeis.org

61, 66, 72, 73, 76, 77, 84, 86, 92

Offset: 1

Robert Price, Nov 27 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. A025431, A287166, A295698.

cp's OEIS Frontend

A287165 Smallest number with exactly n representations as a sum of 6 nonzero squares or 0 if no such number exists.

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A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

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A295797 Numbers that have exactly one representation as a sum of seven positive squares.

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A295799 Numbers that have exactly two representations as a sum of seven positive squares.

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A295800 Numbers that have exactly three representations as a sum of seven positive squares.

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A295801 Numbers that have exactly four representations as a sum of seven positive squares.

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A295802 Numbers that have exactly five representations as a sum of seven positive squares.

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A295803 Numbers that have exactly six representations as a sum of seven positive squares.

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A295804 Numbers that have exactly seven representations as a sum of seven positive squares.

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A295805 Numbers that have exactly eight representations as a sum of seven positive squares.

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