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, 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, 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, 1, 1, 1, 2

Offset: 1

Author

Richard R. Forberg, Sep 22 2013

Keywords

• nonn

Comments

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.

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.

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.

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.

Crossrefs

A024702