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A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

%I A263723 #17 Nov 02 2015 17:09:04
%S A263723 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 A263723 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 A263723 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 A263723 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 A263723 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 A263723 W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see pp. 220-221.
%H A263723 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's Hypothesis H</a>
%e A263723 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 A263723 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 A263723 lst = {}; r = 7; While[r < 132, q = 5; While[q < r, P = 9 + q^2 + r^2; If[PrimeQ@P, AppendTo[lst, P]];
%t A263723   q = NextPrime@q]; r = NextPrime@r]; Take[Transpose[Tally@Sort@lst][[2]], 105]
%Y A263723 Cf. A085317, A125516, A182479.
%K A263723 nonn
%O A263723 1,5
%A A263723 _Jonathan Sondow_ and _Robert G. Wilson v_, Oct 24 2015