%I A061357
%S A061357 0,0,0,1,1,1,1,2,2,2,2,3,2,2,3,2,3,4,1,3,4,3,3,5,4,3,5,3,3,6,2,5,6,2,5,
%T A061357 6,4,5,7,4,4,8,4,4,9,4,4,7,3,6,8,5,5,8,6,7,10,6,5,12,3,5,10,3,7,9,5,5,
%U A061357 8,7,7,11,5,5,12,4,8,11,4,8,10,5,5,13,9,6,11,7,6,14,6,8,13,5,8,11,6,9
%N A061357 Number of 1<=k<n such that n-k and n+k are both primes.
%C A061357 Number of prime pairs (p,q) with p < n < q and q-n = n-p.
%C A061357 The same as the number of ways n can be expressed as the mean of two 
               distinct primes.
%C A061357 Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003
%C A061357 Conjectures from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 24 
               2003: 1) For each integer N>=1 there exists a positive integer m(N) 
               such that for n>=m(N) a(n)>a(N). (After the first m(N)-1 terms, a(N) 
               does not reappear). In particular, for N=1 (or 2 or 3), m(N)=4 and 
               a(N)=0, giving Benoit Cloitre's conjecture. (cont.)
%C A061357 (cont.) Conjectures based upon observing a(1),...,a(10000):
%C A061357 m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1,
%C A061357 m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2,
%C A061357 m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3,
%C A061357 m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4,
%C A061357 m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5,
%C A061357 m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, 
               etc.
%C A061357 2) Each nonnegative integer appears at least once in the current sequence.
%C A061357 3) Stronger than 2): A001477 (nonnegative integers) is a subsequence 
               of the current sequence. (Supporting evidence: I've observed that 
               0,1,2,...,175 is a subsequence of a(1),...,a(10000)).
%C A061357 a(n) is also the number of k such that 2*k+1=p and 2*(n-k-1)+1=q are 
               both odd primes with p < q with p*q = n^2 - m^2 [From Pierre CAMI 
               (pierrecami(AT)tele2.fr), Sep 01 2008]
%C A061357 Also: Number of ways n^2 can be written as b^2+pq where 0<b<n-1 and p,
               q are primes. - Erin Noel and George Panos (erin.m.noel(AT)rice.edu), 
               Jun 27 2006
%H A061357 P. CAMI, <a href="b061357.txt">Table of n, a(n) for n = 4..60000</a>
%F A061357 a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007
%e A061357 a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/
               2 = (7+13)/2.
%p A061357 P:=proc(i) local a,b,c,n; print(0); print(0); print(0); for n from 4 
               by 1 to i do a:=0; b:=prevprime(n); while b>2 do c:=2*n-b; if isprime(c) 
               then a:=a+1; fi; b:=prevprime(b); od; print(a); od; end: P(100); 
               [From Paolo P. Lava (ppl(AT)spl.at), Dec 22 2008]
%Y A061357 Cf. A071681 (subsequence for prime n only).
%Y A061357 Sequence in context: A089993 A047931 A033618 this_sequence A138139 A127992 
               A067595
%Y A061357 Adjacent sequences: A061354 A061355 A061356 this_sequence A061358 A061359 
               A061360
%K A061357 nonn,easy
%O A061357 1,8
%A A061357 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 28 2001
%E A061357 More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001