A006431 -id:A006431 - cp's OEIS frontend

A178419 Partial sums of A006431.

0, 1, 3, 6, 11, 17, 24, 32, 43, 57, 72, 95, 119, 151, 207, 303, 431, 655, 1039, 1551, 2447, 3983, 6031, 9615, 15759, 23951, 38287, 62863, 95631, 152975, 251279, 382351, 611727, 1004943, 1529231, 2446735, 4019599, 6116751, 9786767, 16078223

Offset: 1

Jonathan Vos Post, May 27 2010

Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4).

Cf. A006431.

a(n) = Sum_{i=1..n} A006431(i).

G.f.: -x^2*(36*x^13 +28*x^12 +32*x^11 +21*x^10 +17*x^9 +14*x^8 +13*x^7 +12*x^6 +5*x^5 +2*x^4 -x^3 -3*x^2 -2*x -1) / ((x -1)*(4*x^3 -1)). - Colin Barker, Apr 20 2013

A002635 Number of partitions of n into 4 squares.

Original entry on oeis.org

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

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 *)

A000534 Numbers that are not the sum of 4 nonzero squares.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152, 3670016, 6291456, 8388608, 14680064

Offset: 1

N. J. A. Sloane and J. H. Conway

For n > 15, a(n) = A006431(n-1). - Thomas Ordowski, Nov 18 2012

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 302.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, Theorem 3, pp. 74-75.

T. D. Noe, Table of n, a(n) for n = 1..100
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024. See p. 2.
Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
Index entries for linear recurrences with constant coefficients, signature (0,0,4).
Index entries for sequences related to sums of squares

Cf. A123069, A000414 (complement).

q=22;lst={};Do[Do[Do[Do[z=a^2+b^2+c^2+d^2;If[z<=q^2+3,AppendTo[lst,z]],{d,q}],{c,q}],{b,q}],{a,q}];lst1=Union@lst lst={};Do[AppendTo[lst,n],{n,q^2+3}];lst2=lst Complement[lst2,lst1] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Join[{0,1,2,3,5,6,8,9,11,14,17,24,29,32,41}, LinearRecurrence[{0, 0, 4}, {56, 96, 128}, 30]] (* Jean-François Alcover, Feb 09 2016 *)

for(n=1,224,if(sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,if(a^2+b^2+c^2+d^2-n,0,1)))))==0,print1(n,",")))

{a(n)=if( n<2, 0, n<16, [1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41][n-1], [4, 7, 12][n%3+1] * 2^(n\3*2-7))}; /* Michael Somos, Apr 23 2006 */

is(n)=my(k=if(n,n/4^valuation(n,4),2)); k==2 || k==6 || k==14 || setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014

from itertools import count, islice
def A000534_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2,6,14},count(max(startvalue,0)))
A000534_list = list(islice(A000534_gen(),30)) # Chai Wah Wu, Jul 09 2022

Consists of the numbers 0, 1, 3, 5, 9, 11, 17, 29, 41, 2*4^m, 6*4^m and 14*4^m (m >= 0). Compare A123069.
From 224 on, a(n) = 4*a(n-3).
Numbers n such that A025428(n) = 0.
G.f.: x^2*(36*x^16 + 32*x^15 + 60*x^14 + 55*x^13 + 36*x^12 + 27*x^11 + 20*x^10 + 19*x^9 + 18*x^8 + 13*x^7 + 11*x^6 + 4*x^5 + 2*x^4 - x^3 - 3*x^2 - 2*x - 1)/(4*x^3 - 1). - Chai Wah Wu, Jul 09 2022

A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

4, 9, 10, 12, 13, 16, 17, 19, 20, 21, 22, 29, 30, 31, 35, 39, 40, 44, 46, 47, 48, 64, 71, 80, 88, 120, 160, 176, 184, 192, 256, 320, 352, 480, 640, 704, 736, 768, 1024, 1280, 1408, 1920, 2560, 2816, 2944, 3072, 4096, 5120, 5632, 7680

Offset: 1

Ant King, Aug 17 2010

The largest odd member of this sequence is 71, and from a(32)=320 onwards the terms satisfy the eighth-order recurrence relation a(n)=4a(n-8).

A002635(a(n)) = 2. - Reinhard Zumkeller, Jul 13 2014

			As the fifth integer which has precisely two partitions into sums of four squares of nonnegative numbers is 13, then a(5)=13.

Robert Price, Table of n, a(n) for n = 1..65
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A245022.

a180149 n = a180149_list !! (n-1)
a180149_list = filter ((== 2) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014

Select[Range[10000], Length[PowersRepresentations[ #, 4, 2]]==2&]

The members of this sequence are {9, 13, 17, 19, 21, 29, 30, 31, 35, 39, 46, 47, 71} together with all integers of the form 5*2^N, 11*2^N and {1,3,30,46}*4^N where N > 0 (which includes a necessary correction to Lehmer's result).

A245022 Integers with precisely three partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

18, 25, 26, 27, 28, 33, 37, 38, 41, 43, 51, 53, 55, 59, 60, 62, 72, 79, 92, 95, 104, 112, 152, 240, 248, 288, 368, 416, 448, 608, 960, 992, 1152, 1472, 1664, 1792, 2432, 3840, 3968, 4608, 5888, 6656, 7168, 9728, 15360, 15872, 18432, 23552, 26624, 28672

Offset: 1

Reinhard Zumkeller, Jul 13 2014

nonn

A002635(a(n)) = 3.

			a(1) = 18 = 16 + 1 + 1 + 0 = 9 + 9 + 0 + 0 = 9 + 4 + 4 + 1;
a(2) = 25 = 25 + 0 + 0 + 0 = 16 + 9 + 0 + 0 = 16 + 4 + 4 + 1;
a(3) = 26 = 25 + 1 + 0 + 0 = 16 + 9 + 1 + 0 = 9 + 9 + 4 + 4;
a(4) = 27 = 25 + 1 + 1 + 0 = 16 + 9 + 1 + 1 = 9 + 9 + 9 + 0;
a(5) = 28 = 25 + 1 + 1 + 1 = 16 + 4 + 4 + 4 = 9 + 9 + 9 + 1;
a(6) = 33 = 25 + 4 + 4 + 0 = 16 + 16 + 1 + 0 = 16 + 9 + 4 + 4;
a(7) = 37 = 36 + 1 + 0 + 0 = 25 + 4 + 4 + 4 = 16 + 16 + 4 + 1;
a(8) = 38 = 36 + 1 + 1 + 0 = 25 + 9 + 4 + 0 = 16 + 9 + 9 + 4;
a(9) = 41 = 36 + 4 + 1 + 0 = 25 + 16 + 0 + 0 = 16 + 16 + 9 + 0;
a(10) = 43 = 25 + 16 + 1 + 1 = 25 + 9 + 9 + 0 = 16 + 9 + 9 + 9;
a(11) = 51 = 49 + 1 + 1 + 0 = 25 + 25 + 1 + 0 = 25 + 16 + 9 + 1;
a(12) = 53 = 49 + 4 + 0 + 0 = 36 + 16 + 1 + 0 = 36 + 9 + 4 + 4.

Robert Price, Table of n, a(n) for n = 1..55
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A180149.

a245022 n = a245022_list !! (n-1)
a245022_list = filter ((== 3) . a002635) [0..]

Select[ Range@ 30000, Length@PowersRepresentations[#, 4, 2] == 3 &] (* Robert G. Wilson v, Oct 27 2017 *)

a(44)-a(50) from Robert Price, Oct 26 2017

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 *)

A294282 Integers with precisely four partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

34, 36, 42, 45, 49, 57, 61, 63, 65, 67, 68, 69, 77, 78, 83, 87, 94, 107, 116, 119, 136, 144, 168, 272, 312, 376, 464, 544, 576, 672, 1088, 1248, 1504, 1856, 2176, 2304, 2688, 4352, 4992, 6016, 7424, 8704, 9216, 10752, 17408, 19968, 24064, 29696, 34816, 36864, 43008, 69632, 79872, 96256

Offset: 1

Robert Price, Oct 26 2017

nonn

A002635(a(n)) = 4.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A002635, A006431, A180149, A245022.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==4 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294297 Integers with precisely five partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

50, 52, 54, 58, 70, 73, 74, 75, 76, 84, 85, 86, 89, 91, 93, 101, 103, 109, 111, 113, 127, 131, 140, 142, 143, 151, 167, 191, 200, 208, 216, 232, 280, 296, 304, 336, 344, 560, 568, 800, 832, 864, 928, 1120, 1184, 1216, 1344, 1376, 2240, 2272, 3200, 3328, 3456

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 5.

Robert Price, Table of n, a(n) for n = 1..81
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 3500, f@# == 5 &] (* Robert G. Wilson v, Oct 27 2017 *)

A124978 Smallest positive number which has exactly n different partitions as a sum of 4 squares x^2+y^2+z^2+t^2.

Original entry on oeis.org

1, 4, 18, 34, 50, 66, 82, 114, 90, 130, 150, 178, 162, 198, 318, 210, 250, 234, 322, 406, 465, 330, 306, 402, 462, 390, 474, 378, 490, 486, 654, 610, 522, 450, 778, 678, 642, 570, 666, 726, 594, 714, 770, 774, 986, 630, 738, 945, 1035, 850, 1222, 978, 1014, 918

Offset: 1

Artur Jasinski, Nov 14 2006

nonn

Is it known that a(n) always exists? - Franklin T. Adams-Watters, Dec 18 2006

A002635(a(n)) = n. - Reinhard Zumkeller, Jul 13 2014

			a(4)=34 because 34 is smallest number which has 4 partitions 34=4^2+3^2+3^2+0^2 = 4^2+4^2+1^2+1^2 = 5^2+2^2+2^2+1^2 = 5^2+3^2+0^2+0^2
a(3)=18 which has 3 partitions 18=0^2+0^2+3^2+3^2=0^2+1^2+1^2+4^2=1^2+2^2+2^2+3^2.

Cf. A006431, A094942, A124979-A124983, A000378, A002635, A061262.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a124978 = (+ 1) . fromJust . (`elemIndex` (tail a002635_list))
-- Reinhard Zumkeller, Jul 13 2014

kmin[n_] := If[n<5, 1, 10n](* empirical, should be lowered in case of doubt *);
a[n_] := a[n] = For[k=kmin[n], True, k++, If[Length[PowersRepresentations[ k, 4, 2]] == n, Return[k]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 1000}] (* Jean-François Alcover, Mar 11 2019 *)

cnt4sqr(n)={ local(cnt=0,t2) ; for(x=0,floor(sqrt(n)), for(y=x,floor(sqrt(n-x^2)), for(z=y,floor(n-x^2-y^2), t2=n-x^2-y^2-z^2 ; if( t2>=z^2 && issquare(n-x^2-y^2-z^2), cnt++ ; ) ; ) ; ) ; ) ; return(cnt) ; }
A124978(n)= { local(a=1) ; while(1, if( cnt4sqr(a)==n, return(a) ; ) ; a++ ; ) ; }
{ for(n=1,100, print(n," ",A124978(n)) ; ) ; } \\ R. J. Mathar, Nov 29 2006

Corrected and extended by R. J. Mathar, Nov 29 2006

More terms from Franklin T. Adams-Watters, Dec 18 2006

cp's OEIS Frontend

A178419 Partial sums of A006431.

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A002635 Number of partitions of n into 4 squares.

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

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A000534 Numbers that are not the sum of 4 nonzero squares.

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A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.

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A245022 Integers with precisely three partitions into sums of four squares of nonnegative numbers.

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

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