A000408 -id:A000408 - cp's OEIS frontend

A024795 Numbers that are the sum of 3 nonzero squares, including repetitions.

3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 27, 29, 30, 33, 33, 34, 35, 36, 38, 38, 41, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 51, 53, 54, 54, 54, 56, 57, 57, 59, 59, 61, 62, 62, 65, 66, 66, 66, 67, 68, 69, 69, 70, 72, 73, 74, 74, 75, 75, 76, 77, 77, 78, 81, 81, 81, 82, 83, 83, 84, 86, 86

Offset: 1

Clark Kimberling

nonn

Index entries for sequences related to sums of squares

Cf. A000408.

A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

14, 29, 35, 50, 56, 66, 77, 83, 93, 107, 110, 116, 126, 140, 149, 155, 158, 165, 179, 194, 197, 200, 210, 219, 224, 242, 245, 251, 261, 264, 275, 290, 293, 302, 308, 315, 318, 332, 341, 350, 365, 371, 372, 381, 395, 398, 413, 428, 434, 435, 440, 450, 461, 462, 464, 482

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

			35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.

N:= 1000: # for terms <= N
S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
sort(convert(S,list)); # Robert Israel, Jun 08 2020

for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))

w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))

A347802 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 12, 0, 0, 48, 0, 27, 64, 0, 216, 0, 0, 432, 48, 243, 0, 384, 972, 0, 768, 0, 864, 804, 0, 3456, 600, 0, 0, 1968, 3888, 1350, 3072, 0, 5508, 0, 0, 7776, 2400, 6075, 1728, 9600, 1944, 0, 4096, 7776, 21600, 2022, 0, 3456, 17424, 0, 13824, 21552, 0, 19521, 0, 31104, 15984, 0, 0, 21600, 34896, 11907

Offset: 0

Seiichi Manyama, Sep 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000408, A002102, A037215, A347801, A347803.

a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, (i^2+j^2+k^2==n)*(i*j*k)^2)));

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^3))

a(n) is sum of i^2 * j^2 * k^2 for positive integers i,j,k such that i^2+j^2+k^2=n.

A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

Original entry on oeis.org

1, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3

Offset: 1

Yifan Xie, Apr 05 2023

a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

			For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

Yifan Xie, Table of n, a(n) for n = 1..10000
Wikipedia, Lagrange's four-square theorem
Index entries for sequences related to sums of squares

Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

Cf. A362068 (allows zeros), A362110 (distinct).

findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - Yifan Xie and Thomas Scheuerle, Apr 29 2023

A010002 a(0) = 1, a(n) = 9*n^2 + 2 for n>0.

Original entry on oeis.org

1, 11, 38, 83, 146, 227, 326, 443, 578, 731, 902, 1091, 1298, 1523, 1766, 2027, 2306, 2603, 2918, 3251, 3602, 3971, 4358, 4763, 5186, 5627, 6086, 6563, 7058, 7571, 8102, 8651, 9218, 9803, 10406, 11027, 11666, 12323, 12998, 13691, 14402, 15131, 15878, 16643

Offset: 0

N. J. A. Sloane

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=1, s=2. After 1, all terms are in A000408. [Bruno Berselli, Feb 06 2012]

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2 = 4 can be written as A010008(n+1)^2-a(n+1)*A008588(n+1)^2 = 4. - Vincenzo Librandi, Feb 07 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

[1] cat [9*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 9 Range[43]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {11, 38, 83}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

A010002(n)=9*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

G.f.: (1+x)*(1+7*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*9+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(2)/12 *Pi*coth(Pi/3*sqrt 2) = 1.1606262038.. - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A214328 Intersection of A004214 and A018825.

Original entry on oeis.org

1, 4, 7, 15, 16, 23, 28, 31, 39, 47, 55, 60, 63, 64, 71, 79, 87, 92, 95, 103, 111, 112, 119, 124, 127, 135, 143, 151, 156, 159, 167, 175, 183, 188, 191, 199, 207, 215, 220, 223, 231, 239, 240, 247, 252, 255, 256, 263, 271, 279, 284, 287, 295, 303, 311, 316, 319, 327, 335, 343, 348, 351, 359, 367, 368, 375, 380, 383, 391, 399, 407, 412, 415, 423, 431, 439

Offset: 1

N. J. A. Sloane, Jul 26 2012, following a suggestion from Hans Isdahl, Apr 19 2012

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A004214, A000404, A018825, A214329.

a(n) ~ 6 * n. - Bill McEachen, Mar 24 2024

A214329 Complement of A214328.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109

Offset: 1

N. J. A. Sloane, Jul 26 2012, following a suggestion from Hans Isdahl, Apr 19 2012

nonn

Numbers that are the sum of 2 or 3 nonzero squares. - Altug Alkan, Jan 13 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000404, A000408, A004214, A018825, A214328.

is2(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is3(n) = {my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 13 2016

A231632 Squares that are also sums of 2 and 3 nonzero squares.

Original entry on oeis.org

169, 225, 289, 625, 676, 841, 900, 1156, 1225, 1369, 1521, 1681, 2025, 2500, 2601, 2704, 2809, 3025, 3364, 3600, 3721, 4225, 4624, 4900, 5329, 5476, 5625, 6084, 6724, 7225, 7569, 7921, 8100, 8281, 9025, 9409, 10000, 10201, 10404, 10816, 11025, 11236, 11881, 12100, 12321, 12769, 13225, 13456, 13689, 14161

Offset: 1

Zak Seidov, Nov 12 2013

dead

All terms == {0, 1} (mod 4).
Intersection of A000290, A000404 and A000408.
A square n^2 is the sum of k positive squares for all 1 <= k <= n^2 - 14 iff n^2 is the sum of 2 and 3 positive squares (see A309778 and proof in Kuczma) . Consequently this is a duplicate of A018820. - Bernard Schott, Aug 17 2019

			169 = 13^2 = 5^2 + 12^2 = 3^2 + 4^2 + 12^2;
225 = 15^2 = 9^2 + 12^2 = 2^2 + 5^2 + 14^2.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

Zak Seidov and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..100 from Zak Seidov

Cf. A000290, A000404, A000408, A018820.

A269840 Lesser of twin primes where both are the sum of 3 nonzero squares.

Original entry on oeis.org

17, 41, 59, 107, 137, 179, 227, 281, 347, 419, 521, 569, 617, 641, 659, 809, 827, 857, 881, 1019, 1049, 1091, 1289, 1427, 1451, 1481, 1619, 1667, 1697, 1721, 1787, 1931, 2027, 2081, 2129, 2267, 2339, 2657, 2729, 2801, 2969, 3251, 3257, 3299, 3329, 3371, 3467, 3539

Offset: 1

Altug Alkan, Mar 13 2016

nonn

			17 is a term because 17 = 2^2 + 2^2 + 3^2 and 19 = 1^2 + 3^2 + 3^2.
41 is a term because 41 = 3^2 + 4^2 + 4^2 and 43 = 3^2 + 3^2 + 5^2.
59 is a term because 59 = 3^2 + 5^2 + 5^2 and 61 = 3^2 + 4^2 + 6^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A001359, A085317.

isA000408(n) = my(a, b) ; a=1 ; while(a^2+10, ); p-2}
for(n=1, 1e2, if(isA000408(t(n)) && isA000408(t(n)+2), print1(t(n), ", ")));

A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.

Original entry on oeis.org

25, 100, 400, 1600, 6400, 25600, 102400, 409600, 1638400, 6553600, 26214400, 104857600, 419430400, 1677721600, 6710886400, 26843545600, 107374182400, 429496729600, 1717986918400, 6871947673600, 27487790694400, 109951162777600, 439804651110400, 1759218604441600

Offset: 1

Bernard Schott, Aug 17 2019

This sequence comes from the study of A309778, exactly, A309778(n) = 2 iff n^2 belongs to this sequence here.
According to Draxl link, a(n) is a term of this sequence iff a(n) = 5^2 * 4^(n-1) with n >= 1.
This sequence is a subsequence of A219222 whose terms are all of the form b_0 * 4^k with b_0 in A051952, hence, the only primitive term of this sequence here is 25.

			25 = 5^2 = 3^2 + 4^2,
100 = 10^2 = 6^2 + 8^2,
5^2 * 4^(n-1) = (5 * 2^(n-1))^2 = (3 * 2^(n-1))^2 + (4 * 2^(n-1))^2, but these terms are not the sum of three positive squares.

H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés., Mémoires de la SMF, tome 37 (1974), p. 53-53.
Index entries for linear recurrences with constant coefficients, signature (4).

Intersection of A000290 and A219222.

Cf. A000302, A000378, A000408, A051952, A134422, A309778.

Array[25*4^(# - 1) &, 24] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 25 * 4^(n-1); \\ Jinyuan Wang, Aug 18 2019

a(n) = 5^2 * 4^(n-1) with n >= 1.
a(n) = 4*a(n-1) for n > 1. G.f.: 25*x/(1 - 4*x). - Chai Wah Wu, Aug 29 2019
a(n) = 25 * A000302(n-1). - Alois P. Heinz, Aug 29 2019
E.g.f.: 25*(exp(4*x) - 1)/4. - Stefano Spezia, Oct 28 2023

cp's OEIS Frontend

A024795 Numbers that are the sum of 3 nonzero squares, including repetitions.

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A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

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Maple

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A347802 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^3.

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A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

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A010002 a(0) = 1, a(n) = 9*n^2 + 2 for n>0.

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Magma

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A214328 Intersection of A004214 and A018825.

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A214329 Complement of A214328.

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PARI

A231632 Squares that are also sums of 2 and 3 nonzero squares.

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A269840 Lesser of twin primes where both are the sum of 3 nonzero squares.

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