A294675 -id:A294675 - cp's OEIS frontend

A047700 Numbers that are the sum of 5 positive squares.

5, 8, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Arlin Anderson (starship1(AT)gmail.com)

Complement of A047701.

			From _David A. Corneth_, Aug 04 2020: (Start)
2009 is in the sequence as 2009 = 18^2 + 18^2 + 18^2 + 19^2 + 26^2.
2335 is in the sequence as 2335 = 19^2 + 19^2 + 20^2 + 22^2 + 27^2.
3908 is in the sequence as 3908 = 24^2 + 24^2 + 26^2 + 28^2 + 36^2. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000414, A003328, A047701, A294675, A344795, A344805.

a(n) = n + 12 for n >= 22. - David A. Corneth, Aug 04 2020

A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, May 30 2014

			T(20,5) = 2 = #{ (16,1,1,1,1), (4,4,4,4,4) } since 20 = 4^2 + 4 * 1^2 = 5 * 2^2.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  (...)

Alois P. Heinz, Rows n = 0..200, flattened

Columns k = 0..10 give: A000007, A010052 (for n>0), A025426, A025427, A025428, A025429, A025430, A025431, A025432, A025433, A025434.
Row sums give A001156.
T(2n,n) gives A111178.
T(n^2,n) gives A319435.
Cf. A000290, A276559, A281541, A292520, A341040.
T(n,k) = 1 for n in A025284, A025321, A025357, A294675, A295670, A295797 (for k = 2..7, respectively).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
T:= (n, k)-> b(n, isqrt(n), k):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(s-> `if`(s>n, 0, expand(x*b(n-s, i))))(i^2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, isqrt(n))):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2021

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]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 06 2014, after Alois P. Heinz *)
T[n_, k_] := Count[PowersRepresentations[n, k, 2], r_ /; FreeQ[r, 0]]; T[0, 0] = 1; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2016 *)

T(n,k,L=n)=if(n>k*L^2, 0, k>n-3, k==n, k<2, issquare(n,&n) && n<=L*k, k>n-6, n-k==3, L=min(L,sqrtint(n-k+1)); sum(r=0,min(n\L^2,k-1),T(n-r*L^2,k-r,L-1), n==k*L^2)) \\ M. F. Hasler, Aug 03 2020

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000290(j)).
Sum_{k=1..n} k * T(n,k) = A281541(n).
Sum_{k=1..n} n * T(n,k) = A276559(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A292520(n).

A294524 Numbers that have a unique partition into a sum of five nonnegative squares.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 15

Offset: 1

Robert Price, Nov 01 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 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. A000174, A006431, A294675.

m = 5;
r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, NonNegative] && n == Total[xx^2], xx, Integers];
For[n = 0, n < 20, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* Jean-François Alcover, Feb 25 2019 *)

A294736 Numbers that are the sum of 5 nonzero squares in exactly 2 ways.

Original entry on oeis.org

20, 38, 41, 45, 47, 48, 49, 50, 54, 55, 63, 66, 81, 105

Offset: 1

Robert Price, Nov 07 2017

Inspected values of n <= 50000.

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 two 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." - David A. Corneth, Nov 08 2017

			There are exactly two ways 20 is a sum of 5 nonzero squares. These are 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 = 20. Therefore 20 is in the sequence.

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

Cf. A025429, A025357, A243148, A294675.

Select[Range[200], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 2&] (* Jean-François Alcover, Nov 06 2020 *)

A243148(a(n),5) = 2. - Alois P. Heinz, Feb 26 2019

A295150 Numbers that have exactly two representations as a sum of five nonnegative squares.

Original entry on oeis.org

4, 5, 8, 9, 10, 11, 12, 14, 23, 24

Offset: 1

Robert Price, Nov 15 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, 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. A000174, A006431, A294675.

okQ[n_] := Length[PowersRepresentations[n, 5, 2]] == 2;
Select[Range[100], okQ] (* Jean-François Alcover, Feb 26 2019 *)

A295159 Smallest number with exactly n representations as a sum of five nonnegative squares.

Original entry on oeis.org

0, 4, 13, 20, 29, 37, 50, 52, 61, 74, 77, 85, 91, 101, 106, 118, 125, 131, 133, 139, 162, 157, 154, 166, 178, 194, 187, 205, 203, 202, 227, 211, 226, 235, 234, 269, 251, 275, 250, 266, 291, 274, 259, 283, 301, 325, 306, 298, 326, 334, 347, 322, 362, 447, 331

Offset: 1

Robert Price, Nov 15 2017

nonn

Conjecture: a(448) does not exist, i.e., there is no number with exactly 448 such representations. - Robert Israel, Nov 15 2017

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

Robert Israel, Table of n, a(n) for n = 1..447 (first 200 terms from Robert Price)
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A000174, A006431, A294675.

N:= 1000: # to get a(1)...a(n) where a(n+1) is the first term > N
V:= Array(0..N):
for x[1] from 0 to floor(sqrt(N/5)) do
  for x[2] from x[1] while x[1]^2 + 4*x[2]^2 <= N do
    for x[3] from x[2] while x[1]^2 + x[2]^2 + 3*x[3]^2 <= N do
      for x[4] from x[3] while x[1]^2 + x[2]^2 + x[3]^2 + 2*x[4]^2 <= N do
        for x[5] from x[4] while x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2 <= N do
           t:=  x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2;
           V[t]:= V[t]+1;
od od od od od:
A:= Vector(max(V),-1):
for i from 0 to N do if A[V[i]]=-1 then A[V[i]]:= i fi od:
T:= select(t -> A[t]=-1, [$1..max(V)]):
if T = [] then nmax:= max(V) else nmax:= T[1]-1 fi:
convert(A[1..nmax],list); # Robert Israel, Nov 15 2017

A000174(a(n))=n. - Robert Israel, Nov 15 2017

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

Robert Price, Nov 07 2017

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.

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

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

Robert Price, Nov 07 2017

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.

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

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

Robert Price, Nov 07 2017

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.

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

Cf. A025429, A025357, A294675, A294736.

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

Robert Price, Nov 07 2017

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.

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

Cf. A025429, A025357, A294675, A294736.

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 *)

cp's OEIS Frontend

A047700 Numbers that are the sum of 5 positive squares.

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A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

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A294524 Numbers that have a unique partition into a sum of five nonnegative squares.

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A294736 Numbers that are the sum of 5 nonzero squares in exactly 2 ways.

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A295150 Numbers that have exactly two representations as a sum of five nonnegative squares.

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A295159 Smallest number with exactly n representations as a sum of five nonnegative squares.

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

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

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