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%I #22 Sep 12 2017 09:59:45
%S 0,0,0,1,1,1,1,2,3,3,2,2,3,1,4,4,2,4,4,4,3,5,5,7,9,8,7,8,7,6,7,9,7,9,
%T 8,11,8,8,7,10,9,11,12,9,9,14,11,12,11,15,15,12,14,12,12,17,11,14,15,
%U 15,14,15,18,16,13,18,12,16,14,16,14,12,19,17,13,19
%N a(n) is the number of distinct values that can be expressed as prime(n+k) + prime(n-k) in at least 2 different ways.
%C Conjecture: a(n) > 0 for n > 3.
%H Michel Lagneau, <a href="/A269043/b269043.txt">Table of n, a(n) for n = 1..1000</a>
%e a(13) = 3 because:
%e p(13 + 1) + p(13 - 1) = 43 + 37 = 80;
%e p(13 + 2) + p(13 - 2) = 47 + 31 = 78;
%e p(13 + 3) + p(13 - 3) = 53 + 29 = 82;
%e p(13 + 4) + p(13 - 4) = 59 + 23 = 82;
%e p(13 + 5) + p(13 - 5) = 61 + 19 = 80;
%e p(13 + 6) + p(13 - 6) = 67 + 17 = 84;
%e p(13 + 7) + p(13 - 7) = 71 + 13 = 84;
%e p(13 + 8) + p(13 - 8) = 73 + 11 = 84.
%e p(13 + 9) + p(13 - 9) = 79 + 7 = 86;
%e p(13 + 10) + p(13 - 10) = 83 + 5 = 88;
%e p(13 + 11) + p(13 - 11) = 89 + 3 = 92;
%e p(13 + 12) + p(13 - 12) = 97 + 2 = 99.
%e The 3 distinct values of prime(n+k) + prime(n-k) that are each obtained in at least 2 ways are 80, 82 and 84.
%p for n from 1 to 100 do:
%p lst:={}:W:=array(1..n-1):cr:=0:
%p for m from n-1 by -1 to 1 do:
%p q:=ithprime(n-m)+ithprime(n+m):lst:=lst union {q}:W[m]:=q:
%p od:
%p n0:=nops(lst):c:=0:U:=array(1..n0):
%p for i from 1 to n0 do:
%p c1:=0:
%p for j from 1 to n-1 do:
%p if lst[i]=W[j] then c:=c+1:c1:=c1+1:
%p else fi:
%p od:
%p U[i]:=c1:cr:=cr+1:
%p od:
%p ct:=0:
%p for l from 1 to cr do:
%p if U[l]>1 then ct:=ct+1:
%p else fi:
%p od:
%p printf(`%d, `,ct):
%p od:
%o (PARI) a(n) = {v = []; for (k=1, n-1, v = concat(v, prime(n+k) + prime(n-k));); vd = vecsort(v,,8); sum(k=1, #vd, #select(x->x==vd[k], v)>1);} \\ _Michel Marcus_, Mar 13 2016
%Y Cf. A006562, A055380, A055382.
%K nonn
%O 1,8
%A _Michel Lagneau_, Feb 18 2016
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