cp's OEIS Frontend

A127308 Number of ways of writing the n-th prime prime(n) as a sum of 24 squares.

%I A127308 #25 Aug 08 2023 04:44:51
%S A127308 1104,16192,1362336,44981376,6631997376,41469483552,793229226336,
%T A127308 2697825744960,22063059606912,282507110257440,588326886375936,
%U A127308 4119646755044256,12742799887509216,21517654506205632,57242599902057216
%N A127308 Number of ways of writing the n-th prime prime(n) as a sum of 24 squares.
%C A127308 |a(n) - (16/691)*(prime(n)^11 + 1)| <= (66304/691)*sqrt(prime(n)^11) (proved by Deligne).
%D A127308 E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 107.
%D A127308 Barry Mazur, Controlling our errors, Nature 443, 7 (2006) 38-40.
%H A127308 N. J. A. Sloane, <a href="/A127308/b127308.txt">Table of n, a(n) for n = 1..70</a>
%H A127308 Barry Mazur, <a href="http://www.ams.org/ams/currentevents2007.pdf">The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture</a>, Current Events Bulletin, Amer. Math. Soc., 2007.
%H A127308 Barry Mazur, <a href="http://www.math.harvard.edu/~mazur/papers/nature_sato_tate.pdf">Controlling our errors</a>
%H A127308 Barry Mazur, <a href="https://doi.org/10.1090/S0273-0979-08-01207-X">Finding meaning in error terms</a>, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.
%H A127308 Tony Phillips, <a href="https://web.archive.org/web/20081005194933/http://www.ams.org/mathmedia/archive/10-2006-media.html">Math in the Media</a>
%F A127308 a(n) = A000156(prime(n)).
%F A127308 a(n) ~ (16/691)*(prime(n)^11 + 1) as n -> oo.
%F A127308 a(n) = (16/691)*(prime(n)^11+1) + (33152/691)*tau(prime(n)) for n>1 where tau = A000594. - _Giorgos Kalogeropoulos_, Dec 15 2022
%e A127308 For prime(1) = 2, two of the 24 squares are (+-1)^2 and the other 22 are 0^2, so a(1) = 2*2*binomial(24,2) = 4*276 = 1104.
%t A127308 Table[SquaresR[24, Prime[n]], {n, 1, 70}]
%t A127308 Table[Abs[16/691 (p^11 + 1) + 33152/691 RamanujanTau[p]], {p, Prime@Range@70}] (* _Giorgos Kalogeropoulos_, Dec 15 2022 *)
%Y A127308 Cf. A000040, A000156, A000594.
%K A127308 nonn
%O A127308 1,1
%A A127308 _Jonathan Sondow_, Jan 10 2007