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

1,8

Comments

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

Links

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

Example

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.

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:

Prog

(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

Crossrefs

Cf. A006562, A055380, A055382.

Sequence in context: A048466 A096838 A096007 * A309952 A059252 A349319

Adjacent sequences: A269040 A269041 A269042 * A269044 A269045 A269046

Keyword

nonn

Author

Michel Lagneau, Feb 18 2016

Status

approved