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A222590 Greatest prime representable as the arithmetic mean of two other primes in n different ways, or 0 if no such prime exists.

%I A222590 #23 Feb 27 2018 02:52:32
%S A222590 3,19,31,61,79,83,199,181,229,271,277,313,293,439,389,401,499,619,601,
%T A222590 709,859,643,787,811,743,823,1039,1009,1321,1021,1279,1213,1249,1489,
%U A222590 1483,1301,1609,1621,1459,1753,1559,1877,2011,2029,1741,1901,2087,2239,2207
%N A222590 Greatest prime representable as the arithmetic mean of two other primes in n different ways, or 0 if no such prime exists.
%C A222590 a(6681) is probably the only such term which equals zero.
%H A222590 Robert G. Wilson v, <a href="/A222590/b222590.txt">Table of n, a(n) for n = 0..10000</a>
%H A222590 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A222590 There are only two primes which are not the arithmetic mean of two other primes and they are 2 and 3. Therefore a(0)=3.
%e A222590 There are only three primes which are the arithmetic mean of two other primes in just one way. They are 5 = (3+7)/2, 7 = (3+11)/2, and 19 = (7+31)/2. Therefore a(1)=19.
%e A222590 There are only three primes which are the arithmetic mean of two other primes in just two ways. They are 11 = (3+19)/2 = (5+17)/2, 13 = (3+23)/2 = (7+19)/2, and 31 = (3+59)/2 = (19+43)/2. Therefore a(2)=31, etc.
%t A222590 f[n_] := Block[{c = 0, k = 2, p = Prime@ n}, While[k + 1 < p, If[PrimeQ[p - k] && PrimeQ[p + k], c++ ]; k += 2]; c]; t = Table[0, {1000}]; Do[a = f@ n; If[a < 1001, t[[a + 1]] = Prime@ n; Print[{a, Prime@ n}]], {n, 5000}]; Take[t, 50]
%Y A222590 Cf. A071681, A126204.
%K A222590 nonn
%O A222590 0,1
%A A222590 _Robert G. Wilson v_, Feb 25 2013