A025423 -id:A025423 - 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 *)

A295919 Smallest number with exactly n representations as a sum of eight nonnegative squares.

Original entry on oeis.org

0, 4, 8, 13, 16, 20, 22, 25, 30, 29, 34

Offset: 0

Robert Price, Nov 29 2017

It appears that a(n) does not exist for n in {12, 24, 39, 40, 48, 50, 53, 58, 60, 64, 67, 70, 71, 75, 78, 81, 83, 85, 86, 87, 91, 92, 96, 102, 103, 108, 113, 115, 120, 122, 123, 124, 128, 129, 130, 132, 135, 136, 138, 140, 142, 143, 144, 145, 148, 150, 151, 155, 156, 158, 160, 161, 164, 168, 170, 171, 172, 174, 175, 177, 178, 181, 183, 185, 186, 188, 189, 190, 196, 197, 198, 199, 200}; 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. A025423, A295752.

A295930 Largest number with exactly n representations as a sum of eight nonnegative squares.

Original entry on oeis.org

3, 7, 15, 14, 19, 24, 27, 28, 33, 35, 39

Offset: 1

Robert Price, Nov 29 2017

It appears that a(12) 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. A025423, A025432, A294690.

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A002635 Number of partitions of n into 4 squares.

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A000174 Number of partitions of n into 5 squares.

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A295919 Smallest number with exactly n representations as a sum of eight nonnegative squares.

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A295930 Largest number with exactly n representations as a sum of eight nonnegative squares.

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