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.

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, 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, 15, 14, 15, 18, 16, 13, 18, 12, 16, 14, 16, 14, 12, 19, 17, 13, 19

Offset: 1

Author

Michel Lagneau, Feb 18 2016

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for n > 3.

Examples

			a(13) = 3 because:
p(13 + 1)  + p(13 - 1)  = 43 + 37 = 80;
p(13 + 2)  + p(13 - 2)  = 47 + 31 = 78;
p(13 + 3)  + p(13 - 3)  = 53 + 29 = 82;
p(13 + 4)  + p(13 - 4)  = 59 + 23 = 82;
p(13 + 5)  + p(13 - 5)  = 61 + 19 = 80;
p(13 + 6)  + p(13 - 6)  = 67 + 17 = 84;
p(13 + 7)  + p(13 - 7)  = 71 + 13 = 84;
p(13 + 8)  + p(13 - 8)  = 73 + 11 = 84.
p(13 + 9)  + p(13 - 9)  = 79 + 7  = 86;
p(13 + 10) + p(13 - 10) = 83 + 5  = 88;
p(13 + 11) + p(13 - 11) = 89 + 3  = 92;
p(13 + 12) + p(13 - 12) = 97 + 2  = 99.
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.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006562, A055380, A055382.

Programs

Maple

for n from 1 to 100 do:
  lst:={}:W:=array(1..n-1):cr:=0:
    for m from n-1 by -1 to 1 do:
      q:=ithprime(n-m)+ithprime(n+m):lst:=lst union {q}:W[m]:=q:
    od:
      n0:=nops(lst):c:=0:U:=array(1..n0):
        for i from 1 to n0 do:
         c1:=0:
         for j from 1 to n-1 do:
          if lst[i]=W[j] then c:=c+1:c1:=c1+1:
          else fi:
         od:
        U[i]:=c1:cr:=cr+1:
       od:
       ct:=0:
       for l from 1 to cr do:
       if U[l]>1 then ct:=ct+1:
       else fi:
       od:
       printf(`%d, `,ct):
od:

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