A180149 -id:A180149 - cp's OEIS frontend

A006431 Numbers that have a unique partition into a sum of four nonnegative squares.

0, 1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 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

David M. Bloom

From a(16) = 96 onwards, the terms of this sequence satisfy the third-order recurrence relation a(n) = 4a(n-3). - Ant King, Aug 15 2010

A002635(a(n)) = 1. - Reinhard Zumkeller, Jul 13 2014

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
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
Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Cf. A002635, A180149, A245022.

a006431 n = a006431_list !! (n-1)
a006431_list = filter ((== 1) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014

Select[Range[0,3584], Length[PowersRepresentations[ #,4,2]] == 1&] (* Ant King, Aug 15 2010 *)
CoefficientList[Series[x  (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)/(4 x^3 - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{0,0,4},{0,1,2,3,5,6,7,8,11,14,15,23,24,32,56},50] (* Harvey P. Dale, Nov 26 2015 *)

{a(n)=if(n<2, 0, if(n<15, [1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32] [n-1], [4, 7, 12][n%3+1]*2^(n\3*2-7)))} /* Michael Somos, Apr 23 2006 */

Consists of the seven odd numbers 1, 3, 5, 7, 11, 15, 23, plus 0, and numbers of forms 2*4^k, 6*4^k, 14*4^k, k >= 0.
The set {n nonnegative : A002635(n) = 1}.
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) / (4*x^3 -1). - Colin Barker, Apr 20 2013
log(a(n)) = n*log(4)/3 + C(n) + o(1) where C(n) ~ {-2.82922, -3.00364, -2.90612} for n (mod 3) == {2,0,1}. - Bill McEachen, Oct 21 2022

More terms from James Sellers, Dec 24 1999
Corrected by T. D. Noe, Jun 15 2006
Definition revised by Ant King, May 06 2010
Edited and Grosswald reference added by Wolfdieter Lang, Aug 12 2015

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

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

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 *)

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6

Offset: 0

Marius A. Burtea, Sep 26 2020

a(n) >= 1 for n >= 2 ?.

For n <= 200000, a(n) = 1 only for n = 2, 3, 299, (2 = 1 + 1, 3 = 1 + 2, 299 = 1 + 288) and a(n) = 2 only for n in {4, 5, 35, 59, 79, 95, 97, 149, 169, 179, 389}.

			0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0.
4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2.
15 = 3 + 12 = 5 + 10 = 6 + 9 = 7 + 8 and 3, 5, 6, 7, 8, 9, 10, 12 are in A005349, so a(15) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A005349, A116357, A172151, A172398, A180149, A280829, A319468, A302479, A319395, A337854.

niven:=func; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]];

m = 100; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; Array[a, m, 0] (* Amiram Eldar, Sep 27 2020 *)

A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

66, 81, 97, 99, 105, 110, 115, 121, 123, 124, 137, 139, 141, 149, 155, 156, 158, 159, 164, 179, 188, 239, 264, 284, 440, 496, 624, 632, 656, 752, 1056, 1136, 1760, 1984, 2496, 2528, 2624, 3008, 4224, 4544, 7040, 7936, 9984, 10112, 10496, 12032, 16896, 18176

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 6.

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, A294297.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 19000, f@# == 6 &] (* Robert G. Wilson v, Oct 27 2017 *)

A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

82, 98, 100, 102, 106, 108, 118, 125, 129, 132, 133, 134, 135, 161, 163, 173, 183, 197, 199, 204, 211, 212, 215, 236, 263, 328, 392, 400, 408, 424, 432, 472, 528, 536, 816, 848, 944, 1312, 1568, 1600, 1632, 1696, 1728, 1888, 2112, 2144, 3264, 3392, 3776

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 7.

Robert Price, Table of n, a(n) for n = 1..74
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, A294297, A293175.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@650, f@#==7 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294310 Integers with precisely nine partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

90, 146, 166, 174, 185, 187, 205, 206, 207, 209, 219, 220, 221, 223, 231, 235, 251, 260, 271, 287, 316, 359, 360, 380, 584, 664, 696, 824, 880, 1040, 1264, 1440, 1520, 2336, 2656, 2784, 3296, 3520, 4160, 5056, 5760, 6080, 9344, 10624, 11136, 13184, 14080

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 9.

Robert Price, Table of n, a(n) for n = 1..60
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, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==9 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294309 Integers with precisely eight partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

114, 117, 122, 126, 145, 147, 148, 157, 165, 169, 172, 175, 177, 181, 190, 193, 203, 227, 233, 311, 456, 488, 504, 592, 688, 760, 1824, 1952, 2016, 2368, 2752, 3040, 7296, 7808, 8064, 9472, 11008, 12160, 29184, 31232, 32256, 37888, 44032, 48640

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 8.

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, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==8 &] (* Vincenzo Librandi, Oct 28 2017 *)

cp's OEIS Frontend

A006431 Numbers that have a unique partition into a sum of four nonnegative squares.

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A002635 Number of partitions of n into 4 squares.

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

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

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

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A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

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A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

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A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.