A000223 - OEIS

%I M2619 N1036 #34 May 03 2022 11:06:05

%S 3,7,10,19,32,34,37,51,81,119,122,134,157,160,161,174,221,252,254,294,

%T 305,309,364,371,405,580,682,734,756,763,776,959,1028,1105,1120,1170,

%U 1205,1550,1570,1576,1851,1930,2028,2404,2411,2565,2675,2895,2905,2940,3133,3211,3240,3428

%N 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)).

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seth A. Troisi, <a href="/A000223/b000223.txt">Table of n, a(n) for n = 1..131</a>

%H W. C. Mitchell, <a href="http://dx.doi.org/10.1090/S0025-5718-1966-0195834-3">The number of lattice points in a k-dimensional hypersphere</a>, Math. Comp., 20 (1966), 300-310.

%F a(n) = |A210641(A000092(n))|. - _M. F. Hasler_, Mar 26 2012

%t 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 *)

%o (PARI) m=0;for(n=0,1e4, m<abs(A210641(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

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Revised Jun 28 2005