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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

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