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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.
2
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 18 2016
STATUS
approved