A025431 -id:A025431 - cp's OEIS frontend

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

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).

A287166 Smallest number with exactly n representations as a sum of 7 nonzero squares or 0 if no such number exists.

Original entry on oeis.org

7, 22, 31, 37, 45, 67, 55, 61, 69, 70, 79, 82, 94, 108, 85, 93, 103, 106, 111, 132, 109, 126, 139, 117, 147, 146, 130, 145, 144, 133, 153, 167, 141, 154, 160, 172, 159, 166, 187, 157, 177, 174, 175, 0, 178, 165

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 7 because 7 is the smallest number with exactly 1 representation as a sum of 7 nonzero squares: 7 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 22 because 22 is the smallest number with exactly 2 representations as a sum of 7 nonzero squares: 22 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 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, A025431, A080654, A287165, A287167.

A025431(a(n)) = n for a(n) > 0.

A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 7, 35, 0, 42, 35, 0, 105, 28, 21, 140, 49, 105, 105, 106, 210, 84, 182, 210, 217, 287, 105, 420, 378, 126, 497, 392, 420, 532, 350, 630, 714, 434, 546, 980, 742, 609, 980, 896, 1071, 882, 875, 1470, 1239, 1099, 1155, 1722, 1652, 882, 1933, 1995, 1554, 2072, 1505

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A045849, A063691, A063725, A063730, A340481, A340905, A340915, A340946, A340947.

Column k=7 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_3(x) - 1)^7 / 128, where theta_3() is the Jacobi theta function.

A340998 Number of partitions of n into 7 distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2, 3, 0, 0, 2, 2

Offset: 140

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A025441, A025442, A025443, A025444, A045849, A224982, A340906, A340988, A340999, A341000, A341001.

Column k=7 of A341040.

A294690 Largest number with exactly n representations as a sum of seven nonnegative squares.

Original entry on oeis.org

3, 11, 15, 23, 24, 32, 35, 33, 39, 47, 48, 51, 56, 59, 55, 64, 62, 71, 68, 61, 80, 75, 83, 79, 78, 77, 96, 92, 95

Offset: 1

Robert Price, Nov 27 2017

It appears that a(30) does not exist.

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. A025422, A025431, A295669.

A295795 Largest number with exactly n representations as a sum of seven positive squares.

Original entry on oeis.org

20, 44, 38, 68, 65, 60, 83, 89, 92, 81, 107, 99, 116, 110, 108, 90, 131, 128, 140, 134, 132, 125, 155, 143, 127, 164, 158, 148, 149, 144, 163, 179, 167, 151, 176, 185, 172, 188, 173, 203, 180, 177, 174, 195

Offset: 0

Robert Price, Nov 27 2017

It appears that a(44) does not exist.

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. A025422, A025431, A295669, A295702.

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

A295796 The only integers that cannot be partitioned into a sum of seven positive squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20

Offset: 1

Robert Price, Nov 27 2017

Dubouis, E.; L'Interm. des math., vol. 18, (1911), pp. 55-56, 224-225.
Grosswald, E.; Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), pp.73-74.

Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.

Cf. A025431, A047701, A180968.

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.

cp's OEIS Frontend

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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A287166 Smallest number with exactly n representations as a sum of 7 nonzero squares or 0 if no such number exists.

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A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

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A340998 Number of partitions of n into 7 distinct nonzero squares.

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A294690 Largest number with exactly n representations as a sum of seven nonnegative squares.

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A295795 Largest number with exactly n representations as a sum of seven positive squares.

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

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A295796 The only integers that cannot be partitioned into 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.