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 11 results. Next

A294675 Numbers that are the sum of 5 nonzero squares in exactly 1 way.

Original entry on oeis.org

5, 8, 11, 13, 14, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 34, 36, 39, 42, 57, 60
Offset: 1

Views

Author

Robert Price, Nov 06 2017

Keywords

Comments

The sequence is likely to be finite and complete as the next term, if it exists, is > 50000.
From a proof by David A. Corneth on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than one way to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is 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, A243148.

Programs

  • Mathematica
    Select[Range[100], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 1&] (* 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, 5], {n, 0, 100}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

Formula

A243148(a(n),5) = 1. - Alois P. Heinz, Feb 25 2019

A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

It appears that this sequence has 15416 terms, the last of which is 2243453. - Donovan Johnson, Jan 11 2013
From a(1) = 157 we see that c(n) = (number of ways n is the sum of 5 cubes) coincides with A010057 = characteristic function of cubes, up to n = 156. This sequence lists the numbers n for which c(n) = 2. See A003328 for c(n) > 0 and A048926 for c(n) = 1. - M. F. Hasler, Jan 04 2023

Links

Crossrefs

Cf. A003328 (sums of 5 positive cubes), A025404, A048926 (sum of 5 positive cubes in exactly 1 way), A048930, A294736, A343702, A343705, A344237.

Programs

  • Mathematica
    Select[ Range[ 1000], (test = Length[ Select[ PowersRepresentations[#, 5, 3], And @@ (Positive /@ #)& ] ] == 2; If[test, Print[#]]; test)& ](* Jean-François Alcover, Nov 09 2012 *)
  • PARI
    (waycount(n,numcubes,imax)={if(numcubes==0, !n, sum(i=1,imax, waycount(n-i^3,numcubes-1,i)))}); isA048927(n)=(waycount(n,5,floor(n^(1/3)))==2); \\ Michael B. Porter, Sep 27 2009
  • Python
    def ways (n, left = 5, last = 1):
  a = last; a3 = a**3; c = 0
  while a3 <= n-left+1:
    if left > 1:
       c += ways(n-a3, left-1, a)
    elif a3 == n:
       c += 1
    a += 1; a3 = a**3
  return c
for n in range (1,1000): # to print this sequence
  if ways(n)==2: print(n,end=", ") # in Python2 use, e.g.: print n,
# Minor edits by M. F. Hasler, Jan 04 2023

Extensions

More terms from Walter Hofmann (walterh(AT)gmx.de), Jun 01 2000

A344795 Numbers that are the sum of five squares in two or more ways.

Original entry on oeis.org

20, 29, 32, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101
Offset: 1

Views

Author

Sean A. Irvine, May 28 2021

Keywords

Links

Crossrefs

Cf. A025367, A047700, A294736, A343702, A344796, A344806.

A294737 Numbers that are the sum of 5 nonzero squares in exactly 3 ways.

Original entry on oeis.org

29, 32, 35, 37, 40, 43, 44, 46, 51, 52, 58, 65, 69, 73, 78, 87, 90
Offset: 1

Views

Author

Robert Price, Nov 07 2017

Keywords

Comments

This sequence is likely finite and complete as the next term, if it exists, is > 50000.
From a proof by David A. Corneth on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than eight ways to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is 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.

A294738 Numbers that are the sum of 5 nonzero squares in exactly 4 ways.

Original entry on oeis.org

62, 70, 71, 72, 75, 76, 82, 84, 89, 97, 108, 129, 132
Offset: 1

Views

Author

Robert Price, Nov 07 2017

Keywords

Comments

This sequence is likely finite and complete as the next term, if it exists, is > 50000.
From a proof by David A. Corneth on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than eight ways to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is 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, A294737.

A294739 Numbers that are the sum of 5 nonzero squares in exactly 5 ways.

Original entry on oeis.org

53, 56, 59, 61, 64, 67, 68, 74, 79, 93, 95, 96, 102, 111, 114, 153, 177
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 > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, 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 &] == 5]; Select[ Range@ 200, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

A294740 Numbers that are the sum of 5 nonzero squares in exactly 6 ways.

Original entry on oeis.org

80, 86, 92, 98, 100, 103, 110, 113, 117, 121, 135, 145
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 > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, 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 &] == 6]; Select[Range@200, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

A294741 Numbers that are the sum of 5 nonzero squares in exactly 7 ways.

Original entry on oeis.org

77, 83, 85, 88, 94, 99, 120, 124, 130, 137, 138, 150, 156, 201
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 > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, 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 &] == 7]; Select[Range@250, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

A294742 Numbers that are the sum of 5 nonzero squares in exactly 8 ways.

Original entry on oeis.org

91, 104, 106, 119, 122, 123, 126, 141, 143, 162, 185, 225
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 > 5408 then the number of ways to write n as a sum of 5 squares is at least floor(sqrt(n - 101) / 8) = 9. For n <= 5408, 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 &] == 8]; Select[Range@250, fQ] (* Robert G. Wilson v, Nov 17 2017 *)

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 *)
Showing 1-10 of 11 results. Next

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