The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A227698 a(n) is the number of ways to form the integer n as (p^2 + q^2 - 2)/24, where p and q are primes > 3 (excluding swaps of p and q). 0

%I #16 Jan 01 2014 20:47:19

%S 0,1,1,1,0,1,1,1,1,1,0,1,1,2,0,1,2,0,1,1,0,1,1,2,0,0,2,0,1,1,0,0,0,1,

%T 0,1,2,0,0,1,1,2,0,1,1,0,2,0,0,1,0,1,0,0,1,0,1,1,1,0,0,2,0,1,0,0,0,0,

%U 1,1,1,2

%N a(n) is the number of ways to form the integer n as (p^2 + q^2 - 2)/24, where p and q are primes > 3 (excluding swaps of p and q).

%C Define "density" of a(n) as D(n) = sum(i=1 to n, a(n))/n. Define "average size of a hit" as H(n) = sum(i=1 to n, a(n))/sum(i=1 to n, b(n)) where b(n) = 0 if a(n) = 0, and b(n) = 1 if a(n) > 0. While D(n) declines from a maximum of around 88% at n = 17, to 46.6% at n= 1742, and down to 27.8% at n =~ 40000. Whereas H(n) increases from 1.0 to a maximum of 1.3904 at n = 1742 and then declines slowly to about 1.356 at n =~40000. This shows a strong increase in the "clustering tendency" of these sums onto particular values of n up through n = 1741, and strong persistence of that tendency even as the density declines significantly at large n.

%C The "hit density" of a(n), defined as sum(i=1 to n, b(n))/n, reaches its maximum of 80% at n = 10 and declines to 20.51% at n = 40,000 as it continues to fall almost steadily in that range and likely to continue.

%C a(n) reaches values of 8 at n = 3407, 15392, 18282, 32817, 37337 (for n<=40000), which is the highest value of a(n) in this range.

%C A persistent tendency for more frequent large values of a(n) for n> 40000 is conjectured, with the likelihood that 8 is NOT the maximum value, and the possibility that ever larger values can always be found at higher n.

%Y A024702

%K nonn

%O 1,14

%A _Richard R. Forberg_, Sep 22 2013

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 04:31 EDT 2024. Contains 372807 sequences. (Running on oeis4.)