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A269043 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.

%I A269043 #22 Sep 12 2017 09:59:45
%S A269043 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 A269043 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 A269043 15,14,15,18,16,13,18,12,16,14,16,14,12,19,17,13,19
%N A269043 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 A269043 Conjecture: a(n) > 0 for n > 3.
%H A269043 Michel Lagneau, <a href="/A269043/b269043.txt">Table of n, a(n) for n = 1..1000</a>
%e A269043 a(13) = 3 because:
%e A269043 p(13 + 1)  + p(13 - 1)  = 43 + 37 = 80;
%e A269043 p(13 + 2)  + p(13 - 2)  = 47 + 31 = 78;
%e A269043 p(13 + 3)  + p(13 - 3)  = 53 + 29 = 82;
%e A269043 p(13 + 4)  + p(13 - 4)  = 59 + 23 = 82;
%e A269043 p(13 + 5)  + p(13 - 5)  = 61 + 19 = 80;
%e A269043 p(13 + 6)  + p(13 - 6)  = 67 + 17 = 84;
%e A269043 p(13 + 7)  + p(13 - 7)  = 71 + 13 = 84;
%e A269043 p(13 + 8)  + p(13 - 8)  = 73 + 11 = 84.
%e A269043 p(13 + 9)  + p(13 - 9)  = 79 + 7  = 86;
%e A269043 p(13 + 10) + p(13 - 10) = 83 + 5  = 88;
%e A269043 p(13 + 11) + p(13 - 11) = 89 + 3  = 92;
%e A269043 p(13 + 12) + p(13 - 12) = 97 + 2  = 99.
%e A269043 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 A269043 for n from 1 to 100 do:
%p A269043   lst:={}:W:=array(1..n-1):cr:=0:
%p A269043     for m from n-1 by -1 to 1 do:
%p A269043       q:=ithprime(n-m)+ithprime(n+m):lst:=lst union {q}:W[m]:=q:
%p A269043     od:
%p A269043       n0:=nops(lst):c:=0:U:=array(1..n0):
%p A269043         for i from 1 to n0 do:
%p A269043          c1:=0:
%p A269043          for j from 1 to n-1 do:
%p A269043           if lst[i]=W[j] then c:=c+1:c1:=c1+1:
%p A269043           else fi:
%p A269043          od:
%p A269043         U[i]:=c1:cr:=cr+1:
%p A269043        od:
%p A269043        ct:=0:
%p A269043        for l from 1 to cr do:
%p A269043        if U[l]>1 then ct:=ct+1:
%p A269043        else fi:
%p A269043        od:
%p A269043        printf(`%d, `,ct):
%p A269043 od:
%o A269043 (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 A269043 Cf. A006562, A055380, A055382.
%K A269043 nonn
%O A269043 1,8
%A A269043 _Michel Lagneau_, Feb 18 2016