A025422 -id:A025422 - cp's OEIS frontend

A002635 Number of partitions of n into 4 squares.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 2, 2, 1, 3, 4, 2, 4, 3, 3, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 4, 5, 3, 5, 3, 5, 3, 1, 4, 5, 3, 3, 4, 3, 4, 2, 4, 6, 4, 4, 4, 5, 2, 3, 5, 5, 5, 5, 4, 4, 3, 2, 6, 7, 4, 5, 5, 5, 4, 2, 5, 9, 5, 3, 5, 4, 3, 1, 6, 7, 6, 7, 5, 7, 5, 3, 6, 7, 4

Offset: 0

N. J. A. Sloane

a(A124978(n)) = n; a(A006431(n)) = 1; a(A180149(n)) = 2; a(A245022(n)) = 3. - Reinhard Zumkeller, Jul 13 2014

			1: 1000; 2: 1100; 3:1110; 4: 2000 and 1111; 5: 2100; 6: 2110; 7: 2111; 8: 2200; 9: 3000 and 2210; 10: 3100 and 2211; etc.

G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
E. Grosswald, The Problem of the Uniqueness of Essentially Distinct Representations, in Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Index entries for sequences related to sums of squares

Cf. A000290, A124978, A006431, A180149, A245022.

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A000174 (5), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

a002635 = p (tail a000290_list) 4 where
p ks'@(k:ks) c m = if m == 0 then 1 else
if c == 0 || m < k then 0 else p ks' (c - 1) (m - k) + p ks c m
-- Reinhard Zumkeller, Jul 13 2014

Length[PowersRepresentations[ #, 4, 2]] & /@ Range[0, 107] (* Ant King, Oct 19 2010 *)

for(n=1,100,print1(sum(a=0,n,sum(b=0,a,sum(c=0,b,sum(d=0,c,if(a^2+b^2+c^2+d^2-n,0,1))))),","))

a(n)=local(c=0); if(n>=0, forvec(x=vector(4,k,[0,sqrtint(n)]),c+=norml2(x)==n,1)); c

A000174 Number of partitions of n into 5 squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 3, 3, 3, 4, 3, 3, 2, 2, 4, 4, 4, 4, 5, 3, 3, 4, 3, 5, 5, 5, 6, 5, 3, 5, 5, 5, 6, 5, 6, 5, 4, 4, 6, 7, 6, 8, 8, 7, 5, 6, 5, 8, 8, 4, 9, 7, 6, 7, 7, 8, 9, 9, 7, 9, 6, 7, 8, 10, 9, 9, 11, 7, 8, 8, 8, 11, 11, 9, 12, 11, 7, 9, 9, 12, 13, 9, 10, 11, 8, 6, 10, 13, 13, 13

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A002635 (4), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

Cf. A025429, A295160 (largest number k with a(k) = n).

Table[PowersRepresentations[n, 5, 2] // Length, {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

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.

A295748 Numbers that have exactly seven representations of a sum of seven nonnegative squares.

Original entry on oeis.org

25, 28, 31, 35

Offset: 1

Robert Price, Nov 26 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 and allows two more 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. A025422, A295490.

A295752 Smallest number with exactly n representations as a sum of seven nonnegative squares.

Original entry on oeis.org

0, 4, 9, 13, 18, 21, 25, 29, 34, 36, 37, 46, 49, 50, 45, 53, 58, 54, 68, 61, 66, 74, 69, 70, 78, 77, 81, 84, 86

Offset: 0

Robert Price, Nov 26 2017

nonn

It appears that a(n) does not exist for n in {30, 35, 45, 49, 57, 63, 67, 75, 77, 78, 82, 84, 85, 97, 100, 101, 104, 110, 112, 115, 116, 119, 123, 124, 125, 134, 136, 137, 140, 142, 143, 148, 149, 150, 151, 158, 159, 160, 162, 168, 170, 172, 174, 175, 176, 180, 183, 184, 185, 187, 188, 191, 198}; i.e., there is no integer whose number of representations is any of these values.

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, A295494.

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.

A295742 Numbers that have exactly two representations of a sum of seven nonnegative squares.

Original entry on oeis.org

4, 5, 6, 7, 8, 11

Offset: 1

Robert Price, Nov 26 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 and allows two more 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. A025422, A295485.

A295743 Numbers that have exactly three representations of a sum of seven nonnegative squares.

Original entry on oeis.org

9, 10, 12, 14, 15

Offset: 1

Robert Price, Nov 26 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 and allows two more 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. A025422, A295486.

A295744 Numbers that have exactly four representations of a sum of seven nonnegative squares.

Original entry on oeis.org

13, 16, 17, 19, 23

Offset: 1

Robert Price, Nov 26 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 and allows two more 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. A025422, A295487.

A295745 Numbers that have exactly five representations of a sum of seven nonnegative squares.

Original entry on oeis.org

18, 20, 24

Offset: 1

Robert Price, Nov 26 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 and allows two more 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. A025422, A295488.

cp's OEIS Frontend

A002635 Number of partitions of n into 4 squares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

A000174 Number of partitions of n into 5 squares.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A295748 Numbers that have exactly seven representations of a sum of seven nonnegative squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A295752 Smallest number with exactly n representations as a sum of seven nonnegative squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A295742 Numbers that have exactly two representations of a sum of seven nonnegative squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A295743 Numbers that have exactly three representations of a sum of seven nonnegative squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A295744 Numbers that have exactly four representations of a sum of seven nonnegative squares.

Views

Author

Keywords

Comments

References

Links

Crossrefs