A000092 -id:A000092 - cp's OEIS frontend

A000223 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).

3, 7, 10, 19, 32, 34, 37, 51, 81, 119, 122, 134, 157, 160, 161, 174, 221, 252, 254, 294, 305, 309, 364, 371, 405, 580, 682, 734, 756, 763, 776, 959, 1028, 1105, 1120, 1170, 1205, 1550, 1570, 1576, 1851, 1930, 2028, 2404, 2411, 2565, 2675, 2895, 2905, 2940, 3133, 3211, 3240, 3428

Offset: 1

N. J. A. Sloane

nonn

Record values of (absolute values of) A210641 = A117609-A210639. It appears that the records occur always at positive elements of that sequence. (One could add an initial a(0)=1.) - M. F. Hasler, Mar 26 2012

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).

Seth A. Troisi, Table of n, a(n) for n = 1..131
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000323, A000036, A000092, A000413, A000099.

nmax = 3*10^4; P[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi* n^(3/2)]; record = 0; A000223 = Reap[For[n = 1, n <= nmax, n++, If[(p = Abs[pn = P[n]]) > record, record = p; Print[pn]; Sow[pn]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2016 *)

m=0;for(n=0,1e4, mA210641(n)) & print1(m=A210641(n)",")) /* This would print a negative value in case the record in absolute value occured for A117609(n)<A210639(n), which does not happen for n<10^4. */ \\ M. F. Hasler, Mar 26 2012

a(n) = |A210641(A000092(n))|. - M. F. Hasler, Mar 26 2012

Revised Jun 28 2005

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pin^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

Original entry on oeis.org

1, 7, 19, 57, 81, 251, 437, 691, 739, 1743, 3695, 6619, 8217, 9771, 14771, 15155, 16831, 18805, 26745, 30551, 41755, 46297, 54339, 72359, 86407, 96969, 131059, 344859, 395231, 519963, 607141, 677397, 741509, 893019, 917217, 1288415, 1406811, 1789599, 1827927, 3085785, 3216051, 3444439, 3524869

Offset: 0

N. J. A. Sloane

nonn

The initial value a(0) = 1 corresponds to an initial A000092(0) = 0 which is the index of a record in the sense that the value P(0) = 0 is larger than all preceding values, because there are none. - M. F. Hasler, May 04 2022

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).

W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000323, A000036, A000092, A000099, A000223.

Cf. A117609 (A(n) in name).

P[n_] := (s = Sum[SquaresR[3, k], {k, 0, n}]) - Round[(4/3)*Pi*n^(3/2)]; record = 0; A000092 = Reap[For[n = 0, n <= 10^4, n++, If[(p = Abs[P[n]]) > record, record = p; Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Feb 08 2016, after M. F. Hasler in A000092 *)

a(n) = A117609(A000092(n)), considering A000092(0) = 0. - M. F. Hasler, May 04 2022

Revised Jun 28 2005

a(37)-a(42) from Vincenzo Librandi, Aug 21 2016

A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

1, 7, 19, 27, 33, 57, 81, 81, 93, 123, 147, 171, 179, 203, 251, 251, 257, 305, 341, 365, 389, 437, 461, 461, 485, 515, 587, 619, 619, 691, 739, 739, 751, 799, 847, 895, 925, 949, 1021, 1021, 1045, 1141, 1189, 1213, 1237, 1309, 1357, 1357, 1365, 1419, 1503

Offset: 0

John L. Drost, Apr 06 2006

nonn

			a(2) = 1 + 6 + 12 = 19, since (0,0,0) and (0, 0, +-1) and cyclic permutations (for a total of 6 points), and +-(0, 1, +-1) and cyclic permutations (for a total 12 points) are inside or on x^2 + y^2 + z^2 = 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

Partial sums of A005875.
Cf. A000605 (number of points of norm <= n in cubic lattice).
Cf. A210639, A000092 and references therein.
Cf. A057655.

Table[Sum[SquaresR[3,k], {k,0,n}], {n,0,50}] (* T. D. Noe, Apr 08 2006, revised Sep 27 2011 *)

A117609(n)=sum(x=0,sqrtint(n),(sum(y=1,sqrtint(t=n-x^2),1+2*sqrtint(t-y^2))*2+sqrtint(t)*2+1)*2^(x>0)) \\ M. F. Hasler, Mar 26 2012

q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)) /* Joerg Arndt, Apr 08 2013 */

# uses Python code for A057655
from math import isqrt
def A117609(n): return A057655(n)+(sum(A057655(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n) ~ (4/3)*Pi*n^(3/2) ~ A210639(n).
a(n) = A122510(3,n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
a(n^2) = A000605(n). - R. J. Mathar, Aug 03 2025

A000036 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives closest integer to P(A000099(n)).

Original entry on oeis.org

2, 3, 5, 6, 6, -6, 7, 8, 10, 13, 13, 13, 14, -17, 17, 17, 18, -19, 20, -22, 23, 27, -29, -29, 29, -31, -32, -35, 36, -37, -40, -43, -46, -48, -50, -53, -55, -57, -60, -60, -61, -63, -66, -66, -68, -71, -74, -77, -79, -82, -85, -88, -89, -92, -95, -96, -97, -97, -100

Offset: 1

N. J. A. Sloane

sign

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).

David W. Wilson, Table of n, a(n) for n = 1..200
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000092, A000099, A000223, A000323, A000413.

nmax = 6*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]] + 1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000036 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[pn = P[n]]; If[p > record, record = p; k++; Sow[pn // Round]; Print["a(", k, ") = ", pn // Round]]]][[2, 1]] (* Jean-François Alcover, Feb 03 2016 *)

a(n) = round(P(A000099(n))), where P(n) = A057655(n)-pi*n. - David W. Wilson, May 15 2008

Revised by N. J. A. Sloane, Jun 26 2005

More terms from David W. Wilson, May 15 2008

A000099 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

Original entry on oeis.org

1, 2, 5, 10, 20, 24, 26, 41, 53, 130, 149, 205, 234, 287, 340, 410, 425, 480, 586, 840, 850, 986, 1680, 1843, 2260, 2591, 3023, 3024, 3400, 3959, 3960, 5182, 5183, 7920, 9796, 11233, 14883, 15119, 15120, 19593, 21600, 21603, 21604, 22177, 28559, 28560

Offset: 1

N. J. A. Sloane

nonn

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).

David W. Wilson, Table of n, a(n) for n = 1..200
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000323, A000036, A000092, A000413, A000223.

nmax = 3*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]]+1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000099 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[P[n]]; If[p > record, record = p; k++; Sow[n]; Print["a(", k, ") = ", n];]]][[2, 1]] (* Jean-François Alcover, Feb 03 2016 *)

Entry revised by N. J. A. Sloane, Jun 26 2005

A000323 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives A(A000099(n)).

Original entry on oeis.org

5, 9, 21, 37, 69, 69, 89, 137, 177, 421, 481, 657, 749, 885, 1085, 1305, 1353, 1489, 1861, 2617, 2693, 3125, 5249, 5761, 7129, 8109, 9465, 9465, 10717, 12401, 12401, 16237, 16237, 24833, 30725, 35237, 46701, 47441, 47441, 61493, 67797, 67805, 67805

Offset: 1

N. J. A. Sloane

nonn

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).

David W. Wilson, Table of n, a(n) for n = 1..200
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000099, A000036, A000092, A000413, A000223.

nmax = 3*10^4; A[n_] := 1 + 4*Floor[Sqrt[n]] + 4*Floor[Sqrt[n/2]]^2 + 8* Sum[Floor[Sqrt[n - j^2]], {j, Floor[Sqrt[n/2]] + 1, Floor[Sqrt[n]]}]; V[n_] := Pi*n; P[n_] := A[n] - V[n]; record = 0; A000099 = Reap[For[k = 0; n = 1, n <= nmax, n++, p = Abs[P[n]]; If[p > record, record = p; k++; Sow[an = A[n]]; Print["a(", k, ") = ", an];]]][[2, 1]] (* Jean-François Alcover, Feb 07 2016*)

Entry revised Jun 28 2005

A210639 Nearest integer to (4/3)Pin^(3/2).

Original entry on oeis.org

0, 4, 12, 22, 34, 47, 62, 78, 95, 113, 132, 153, 174, 196, 219, 243, 268, 294, 320, 347, 375, 403, 432, 462, 492, 524, 555, 588, 621, 654, 688, 723, 758, 794, 830, 867, 905, 943, 981, 1020, 1060, 1100, 1140, 1181, 1223, 1264, 1307, 1350, 1393, 1437, 1481

Offset: 0

M. F. Hasler, Mar 26 2012

nonn

Approximates the volume of the ball { (x,y,z) | x^2+y^2+z^2 < n }. Provides a more refined scale than A002101(n) = a(n^2).

Cf. A000092 and references therein.

Table[Round[4/3*Pi* n^(3/2)],{n,0,50}] (* Harvey P. Dale, Aug 04 2020 *)

PARI
```
a(n)=round(4/3*Pi*n^1.5)
```

A210641 A117609(n)-A210639(n): Difference between number of lattice points in the ball x^2+y^2+z^2 <= n and the volume of this ball rounded to the nearest integer.

Original entry on oeis.org

1, 3, 7, 5, -1, 10, 19, 3, -2, 10, 15, 18, 5, 7, 32, 8, -11, 11, 21, 18, 14, 34, 29, -1, -7, -9, 32, 31, -2, 37, 51, 16, -7, 5, 17, 28, 20, 6, 40, 1, -15, 41, 49, 32, 14, 45, 50, 7, -28, -18, 22, 25, 4, 31, 81, 34, 36, 36, 13, 37, -12, 11, 58, 8, -36, 10, 55

Offset: 0

M. F. Hasler, Mar 26 2012

sign

Record values are listed in A000223, and A000092 gives the corresponding indices. Strictly speaking, these are defined using the absolute values, but it appears they always occur at positive elements.

a[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi*n^(3/2)]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

PARI
```
A210641(n)=A117609(n)-A210639(n)
```

cp's OEIS Frontend

A000223 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).

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A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)*Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

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A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

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A000036 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives closest integer to P(A000099(n)).

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A000099 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

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A000323 Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives A(A000099(n)).

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A210639 Nearest integer to (4/3)*Pi*n^(3/2).

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A210641 A117609(n)-A210639(n): Difference between number of lattice points in the ball x^2+y^2+z^2 <= n and the volume of this ball rounded to the nearest integer.

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pin^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

A210639 Nearest integer to (4/3)Pin^(3/2).