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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

References

• 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).

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

Revised Jun 28 2005