A263723 - OEIS

%I #17 Nov 02 2015 17:09:04

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

%T 3,2,2,1,1,1,1,1,1,1,2,2,1,1,2,1,1,3,1,1,1,2,2,2,1,1,1,2,1,1,1,2,1,2,

%U 1,1,2,1,1,1,4,1,1,1,1,4,2,2,1,2,1,2,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1

%N Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

%C According to Sierpinski and Schinzel (1988), it is easy to prove that the smallest of p, q, r is always p = 3, and under Schinzel's hypothesis H the sequence is infinite.

%D W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see pp. 220-221.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's Hypothesis H</a>

%e A182479(1) = 83 = 3^2 + 5^2 + 7^2 and A182479(2) = 179 = 3^2 + 7^2 + 11^2 are the only ways to write 83 and 179 as sums of squares of 3 distinct primes, so a(1) = 1 and a(2) = 1.

%e A182479(5) = 419 = 3^2 + 7^2 + 19^2 = 3^2 + 11^2 + 17^2 are the only such representations of 419, so a(5) = 2.

%t lst = {}; r = 7; While[r < 132, q = 5; While[q < r, P = 9 + q^2 + r^2; If[PrimeQ@P, AppendTo[lst, P]];

%t   q = NextPrime@q]; r = NextPrime@r]; Take[Transpose[Tally@Sort@lst][[2]], 105]

%Y Cf. A085317, A125516, A182479.

%K nonn

%O 1,5

%A _Jonathan Sondow_ and _Robert G. Wilson v_, Oct 24 2015