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.

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A294743 Numbers that are the sum of 5 nonzero squares in exactly 9 ways.

Original entry on oeis.org

101, 112, 115, 118, 127, 144, 159, 161, 165, 169, 180
Offset: 1

Views

Author

Robert Price, Nov 07 2017

Keywords

Comments

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 6501 then the number of ways to write n as a sum of 5 squares is at least 10. For n <= 6501 terms have been verified by inspection. Hence this sequence is finite and complete.

References

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

Links

Crossrefs

Cf. A025429, A025357, A294675, A294736.

Programs

  • Mathematica
    fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 9]; Select[Range@250, fQ](* Robert G. Wilson v, Nov 17 2017 *)

A294744 Numbers that are the sum of 5 nonzero squares in exactly 10 ways.

Original entry on oeis.org

107, 109, 116, 125, 140, 146, 168, 209, 249, 273, 297
Offset: 1

Views

Author

Robert Price, Nov 07 2017

Keywords

Comments

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that if n > 7845 then the number of ways to write n as a sum of 5 squares is at least 11. For n <= 7845 terms have been verified by inspection. Hence this sequence is finite and complete.

References

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

Links

Crossrefs

Cf. A025429, A025357, A294675, A294736.

Programs

  • Mathematica
    fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 10]; Select[ Range@300, fQ](* Robert G. Wilson v, Nov 17 2017 *)

A179015 Number of ways in which n^2 can be expressed as the sum of exactly five positive squares.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 2, 6, 6, 9, 9, 15, 8, 25, 20, 21, 25, 39, 26, 46, 44, 57, 49, 71, 52, 102, 81, 81, 99, 145, 92, 156, 126, 164, 160, 204, 151, 247, 217, 236, 245, 326, 211, 357, 319, 381, 360, 416, 344, 518, 446, 476, 450, 670, 468, 675, 607, 661, 668, 825, 625
Offset: 1

Views

Author

Carmine Suriano, Jun 24 2010

Keywords

Comments

As n goes to infinity the ratio of a(n)/a(n) of sequence A178898 (using all different squares) tends to 5/4.

Links

Crossrefs

Cf. A000290, A178898.
Cf. A000132. - R. J. Mathar, Jun 26 2010
Cf. A065458, A065459.

Programs

  • Maple
    a(8) = 5 since 64 can be expressed in five different ways as the sum of 5 squares (order is ignored): 8^2 = 7^2+3^2+2^2+1^2+1^2 = 6^2+5^2+1^2+1^2+1^2 = 6^2+4^2+2^2+2^2+2^2 = 6^2+3^2+3^2+3^2+1^2 = 5^2+5^2+3^2+1^2+1^2.

Formula

Asymptotic behavior for large values of n is a(n) = n^2/2-47n/2+243.
a(n) = A025429(n^2). - R. J. Mathar, Jun 26 2010
a(n) = A065459(n) - A065458(n). - Alois P. Heinz, Oct 25 2018
Previous Showing 21-23 of 23 results.

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

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