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A240301 a(n) is the number of sets of three positive integers p_1 < p_2 < p_3 such that 2*p_2 = p_1 + p_3, where p_i (i=1,2,3) is either 1 or a prime number and p_3 = prime(n).

%I A240301 #23 May 09 2014 22:39:56
%S A240301 1,1,1,1,1,1,2,2,2,2,1,2,3,1,3,3,2,2,4,3,4,4,4,2,3,4,3,3,5,5,5,4,5,4,
%T A240301 5,4,5,5,6,4,4,7,4,6,7,6,7,5,4,5,4,6,8,7,7,7,7,4,8,9,8,5,9,6,7,8,4,8,
%U A240301 8,10,8,6,6,10,9,9,7,7,6,9,10,9,8,8,12,13
%N A240301 a(n) is the number of sets of three positive integers p_1 < p_2 < p_3 such that 2*p_2 = p_1 + p_3, where p_i (i=1,2,3) is either 1 or a prime number and p_3 = prime(n).
%C A240301 a(n)>0 for n > 1.
%C A240301 It is conjectured that every positive integer appears a positive finite number of times in this sequence.
%C A240301 The sequence of records is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 27, 28, 31, 33, 34, 35, 36, 39, 40, 41, 42, 43, 46, 47, 48, 50, 51, 53, 55, 56, 58, 61, 62, 64, 65, 66, 70, 71, 72, 74, 76, 78,... - _R. J. Mathar_, May 02 2014
%C A240301 Alternative definition for p_i is p_1 is either 1 or an odd prime, p_2 is an odd prime after a(2) and p_3 is prime(n). - _Jon Perry_, Apr 17 2014.
%H A240301 Lei Zhou, <a href="/A240301/b240301.txt">Table of n, a(n) for n = 2..10001</a>
%e A240301 For n=2, p_3=prime(2)=3, 2*2=1+3. One instance found, so a(2)=1;
%e A240301 ...
%e A240301 For n=8, p_3=prime(8)=19, 2*11=3+19, 2*13=7+19. Two instances found, so a(8)=2;
%e A240301 ...
%e A240301 For n=30, p_3=prime(30)=113, 2*59=5+113, 2*71=29+113, 2*83=53+113, 2*101=89+113, 2*107=101+113. Five instances found, so a(30)=5.
%t A240301 Table[p = Prime[n]; ct = 0; pp = p; While[pp = NextPrime[pp, -1]; diff = p - pp; diff < pp, cp = pp - diff; If[(PrimeQ[cp]) || (cp == 1), ct++]]; ct, {n, 2, 87}]
%Y A240301 Cf. A000040, A240232.
%K A240301 nonn,easy
%O A240301 2,7
%A A240301 _Lei Zhou_, Apr 03 2014