A063725 -id:A063725 - cp's OEIS frontend

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of ways to write n as an ordered sum of two positive cubes.

			a(9) = 2 because we have [8, 1] and [1, 8].

Antti Karttunen, Table of n, a(n) for n = 0..65537
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Cf. A000578, A001235 (positions of terms > 3), A003325 (of nonzero terms), A010057, A063725, A173677.

nmax = 150; CoefficientList[Series[(Sum[x^(k^3), {k, 1, nmax}])^2, {x, 0, nmax}], x]

A010057(n) = ispower(n, 3);
A280618(n) = if(n<2, 0, sum(r=1,sqrtnint(n-1,3),A010057(n-(r^3)))); \\ Antti Karttunen, Nov 30 2021

G.f.: (Sum_{k>=1} x^(k^3))^2.

A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 6, 20, 0, 30, 15, 0, 60, 12, 15, 60, 31, 60, 30, 60, 90, 36, 86, 60, 120, 120, 15, 180, 141, 60, 165, 140, 180, 186, 120, 180, 285, 156, 126, 360, 255, 216, 270, 260, 390, 240, 262, 420, 426, 360, 210, 540, 530, 216, 540, 540, 480, 600, 300, 600, 825, 312, 576, 840

Offset: 6

Ilya Gutkovskiy, Jan 31 2021

Alois P. Heinz, Table of n, a(n) for n = 6..20000
Index entries for sequences related to sums of squares

Cf. A000141, A000290, A010052, A025430, A045848, A063691, A063725, A063730, A340481, A340906, A340915, A340946, A340947.

Column k=6 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..71);  # Alois P. Heinz, Jan 31 2021

nmax = 71; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (theta_3(x) - 1)^6 / 64, where theta_3() is the Jacobi theta function.

A340946 Number of ways to write n as an ordered sum of 9 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 9, 0, 0, 36, 0, 9, 84, 0, 72, 126, 0, 252, 135, 36, 504, 156, 252, 630, 288, 756, 576, 606, 1260, 756, 1207, 1260, 1584, 2052, 1008, 2727, 2688, 1764, 3663, 2718, 3816, 4608, 2853, 5418, 6048, 4620, 5868, 7506, 7464, 7308, 8442, 8958, 11088, 10404, 9684, 13986, 14184, 13020

Offset: 9

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008452, A010052, A025433, A045851, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340947.

Column k=9 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..63);  # Alois P. Heinz, Jan 31 2021

nmax = 63; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^9/512, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (theta_3(x) - 1)^9 / 512, where theta_3() is the Jacobi theta function.

A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 12, 30, 68, 144, 289, 556, 1034, 1868, 3292, 5678, 9608, 15984, 26188, 42314, 67509, 106460, 166090, 256552, 392628, 595696, 896484, 1338894, 1985298, 2923840, 4278448, 6222518, 8997544, 12938368, 18507297, 26340040, 37307326, 52597320, 73825504, 103180702

Offset: 2

Ilya Gutkovskiy, Jan 30 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..37);  # Alois P. Heinz, Feb 10 2021

nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
A015128[n_] := Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; a[n_] := (1/4) Sum[A015128[k] A015128[n - k], {k, 1, n - 1}]; Table[a[n], {n, 2, 37}]

G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.
a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).
a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).
a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.

A340481 Number of ways to write n as an ordered sum of 5 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 5, 0, 0, 10, 0, 5, 10, 0, 20, 5, 0, 30, 6, 10, 20, 20, 30, 5, 30, 30, 20, 35, 10, 60, 45, 0, 60, 50, 30, 45, 50, 60, 70, 35, 30, 110, 50, 31, 110, 80, 80, 50, 70, 120, 70, 75, 90, 140, 110, 20, 140, 160, 60, 135, 120, 120, 180, 40, 130, 230, 80, 120, 170, 200, 155, 85, 200, 190

Offset: 5

Ilya Gutkovskiy, Jan 31 2021

Alois P. Heinz, Table of n, a(n) for n = 5..20000
Index entries for sequences related to sums of squares

Cf. A000132, A000290, A010052, A025429, A038671, A063691, A063725, A063730, A340905, A340906, A340915, A340946, A340947.

Column k=5 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..75);  # Alois P. Heinz, Jan 31 2021

nmax = 75; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^5/32, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (theta_3(x) - 1)^5 / 32, where theta_3() is the Jacobi theta function.

A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 7, 35, 0, 42, 35, 0, 105, 28, 21, 140, 49, 105, 105, 106, 210, 84, 182, 210, 217, 287, 105, 420, 378, 126, 497, 392, 420, 532, 350, 630, 714, 434, 546, 980, 742, 609, 980, 896, 1071, 882, 875, 1470, 1239, 1099, 1155, 1722, 1652, 882, 1933, 1995, 1554, 2072, 1505

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..20000
Index entries for sequences related to sums of squares

Cf. A000290, A008451, A010052, A025431, A045849, A063691, A063725, A063730, A340481, A340905, A340915, A340946, A340947.

Column k=7 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_3(x) - 1)^7 / 128, where theta_3() is the Jacobi theta function.

A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 8, 0, 0, 28, 0, 8, 56, 0, 56, 70, 0, 168, 64, 28, 280, 84, 168, 280, 176, 420, 224, 345, 560, 392, 616, 420, 848, 924, 336, 1246, 1064, 868, 1464, 988, 1680, 1820, 1120, 1904, 2464, 1932, 1904, 2870, 2752, 2772, 2912, 2892, 4256, 3640, 3248, 4480, 5040, 4760, 3696, 6120

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..20000
Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A045850, A063691, A063725, A063730, A340481, A340905, A340906, A340946, A340947.

Column k=8 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..64);  # Alois P. Heinz, Jan 31 2021

nmax = 64; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_3(x) - 1)^8 / 256, where theta_3() is the Jacobi theta function.

A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 10, 0, 0, 45, 0, 10, 120, 0, 90, 210, 0, 360, 262, 45, 840, 300, 360, 1260, 480, 1260, 1350, 1015, 2520, 1560, 2200, 3150, 2880, 4186, 2880, 5430, 6240, 3780, 8300, 7080, 7920, 11160, 7320, 13257, 14640, 10600, 16470, 18570, 18240, 19620, 22230, 25135, 27720, 28020, 28480, 38160

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

David A. Corneth, Table of n, a(n) for n = 10..10009
Index entries for sequences related to sums of squares

Cf. A000144, A000290, A010052, A025434, A045852, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340946.

Column k=10 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^10/1024, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (theta_3(x) - 1)^10 / 1024, where theta_3() is the Jacobi theta function.

A081324 Twice a square but not the sum of 2 distinct squares.

Original entry on oeis.org

0, 2, 8, 18, 32, 72, 98, 128, 162, 242, 288, 392, 512, 648, 722, 882, 968, 1058, 1152, 1458, 1568, 1922, 2048, 2178, 2592, 2888, 3528, 3698, 3872, 4232, 4418, 4608, 4802, 5832, 6272, 6498, 6962, 7688, 7938, 8192, 8712, 8978, 9522, 10082, 10368, 11552

Offset: 1

Benoit Cloitre, Apr 20 2003

nonn

Conjecture: for n>1 this is A050804.
From Altug Alkan, Apr 12 2016: (Start)
Conjecture is true. Proof :
If n = a^2 + b^2, where a and b are nonzero integers, then n^3 = (a^2 + b^2)^3 = A^2 + B^2 = C^2 + D^2 where;
A = 2*a^2*b + (a^2-b^2)*b = 3*a^2*b - b^3,
B = 2*a*b^2 - (a^2-b^2)*a = 3*a*b^2 - a^3,
C = 2*a*b^2 + (a^2-b^2)*a = 1*a*b^2 + a^3,
D = 2*a^2*b - (a^2-b^2)*b = 1*a^2*b + b^3.
Obviously, A, B, C, D are always nonzero because a and b are nonzero integers. Additionally, if a^2 is not equal to b^2, then (A, B) and (C, D) are distinct pairs, that is, n^3 can be expressible as a sum of two nonzero squares more than one way. Since we know that n is a sum of two nonzero squares if and only if n^3 is a sum of two nonzero squares (see comment section of A000404); if n^3 is the sum of two nonzero squares in exactly one way, n must be a^2 + b^2 with a^2 = b^2 and n is the sum of two nonzero squares in exactly one way. That is the definition of this sequence, so this sequence is exactly A050804 except "0" that is the first term of this sequence. (End) [Edited by Altug Alkan, May 14 2016]
Conjecture: sequence consists of numbers of form 2*k^2 such that sigma(2*k^2)==3 (mod 4) and k is not divisible by 5.
The reason of related observation is that 5 is the least prime of the form 4*m+1. However, counterexamples can be produced. For example 57122 = 2*169^2 and sigma(57122) == 3 (mod 4) and it is not divisible by 5. - Altug Alkan, Jun 10 2016
For n > 0, this sequence lists numbers n such that n is the sum of two nonzero squares while n^2 is not. - Altug Alkan, Apr 11 2016
2*k^2 where k has no prime factor == 1 (mod 4). - Robert Israel, Jun 10 2016

Evan M. Bailey, Table of n, a(n) for n = 1..20000 (first 115 terms from Reinhard Zumkeller, terms 116-1000 from Zak Seidov)
Evan M. Bailey, a081324.cpp

Cf. A050804, A025284, A063725, A125022, A018825, A000404, A004431.

import Data.List (elemIndices)
a081324 n = a081324_list !! (n-1)
a081324_list = 0 : elemIndices 1 a063725_list
-- Reinhard Zumkeller, Aug 17 2011

map(k -> 2*k^2, select(k -> andmap(t -> t[1] mod 4 <> 1, ifactors(k)[2]), [$0..100])); # Robert Israel, Jun 10 2016

Select[ Range[0, 12000], MatchQ[ PowersRepresentations[#, 2, 2], {{n_, n_}}] &] (* Jean-François Alcover, Jun 18 2013 *)

concat([0,2],apply(n->2*n^2, select(n->vecmin(factor(n)[, 1]%4)>1, vector(100,n,n+1)))) \\ Charles R Greathouse IV, Jun 18 2013

A063725(a(n)) = 1. [Reinhard Zumkeller, Aug 17 2011]

a(n) = 2*A004144(n-1)^2 for n > 1. - Charles R Greathouse IV, Jun 18 2013

a(19)-a(45) from Donovan Johnson, Nov 15 2009

Offset corrected by Reinhard Zumkeller, Aug 17 2011

A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

Original entry on oeis.org

0, 2, 5, 50, 65, 1250, 325, 31250, 1105, 8450, 8125, 19531250, 5525, 488281250, 105625, 211250, 27625, 305175781250, 71825, 7629394531250, 138125, 5281250, 126953125, 4768371582031250, 160225, 35701250, 1221025, 2442050, 3453125

Offset: 0

Felix Fröhlich, Oct 05 2019

a(n) is the smallest nonnegative i such that A063725(i) = n.

If a(n) exists, then a(n) is of the form 2*m^2 if and only if n is odd. - Chai Wah Wu, Jun 28 2024

			For n = 3: The sums of the two members of each of the pairs (1, 49), (25, 25) and (49, 1) is 50 and 50 is the smallest nonnegative integer where exactly 3 such pairs exist, so a(3) = 50.

Chai Wah Wu, terms a(n) for n odd and a(n) <= 3433037534088061250.

Cf. A063725, A093195.

a063725(n) = if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ after Charles R Greathouse IV in A063725
a(n) = for(x=0, oo, if(a063725(x)==n, return(x)))

# uses Python code from A063725
from itertools import count
def A328151(n): return next(m for m in ((k**2<<1) if n&1 else k for k in count(0)) if A063725(m)==n) # Chai Wah Wu, Jun 28 2024

Conjecture: a(2k) = A093195(k) for k >= 1, a(2k+1) = 2*A006339(k)^2 for k >= 0. - Jon E. Schoenfield, Jan 23 2022

a(13)-a(22) from Bert Dobbelaere, Oct 20 2019

a(23)-a(28) from Chai Wah Wu, Jun 28 2024

cp's OEIS Frontend

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

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A340905 Number of ways to write n as an ordered sum of 6 squares of positive integers.

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A340946 Number of ways to write n as an ordered sum of 9 squares of positive integers.

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A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

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A340481 Number of ways to write n as an ordered sum of 5 squares of positive integers.

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A340906 Number of ways to write n as an ordered sum of 7 squares of positive integers.

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A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

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A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

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