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 A061357 Number of 0
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Number of prime pairs (p,q) with p < n < q and q-n = n-p. The same as the number of ways n can be expressed as the mean of two distinct primes. Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003 Conjectures from Rick L. Shepherd, Jun 24 2003: (Start) 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.) (cont.) Conjectures based upon observing a(1),...,a(10000): m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1, m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2, m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3, m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4, m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5, m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, etc. 2) Each nonnegative integer appears at least once in the current sequence. 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)). (End) 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. [Pierre CAMI, Sep 01 2008] Also: Number of ways n^2 can be written as b^2+pq where 02 do c:=2*n-b; if isprime(c) then a:=a+1; fi; b:=prevprime(b); od; print(a); od; end: P(100); # Paolo P. Lava, Dec 22 2008 MATHEMATICA Table[Count[Range[n - 1], k_ /; And[PrimeQ[n - k], PrimeQ[n + k]]], {n, 98}] (* Michael De Vlieger, May 14 2016 *) PROG (Haskell) a061357 n = sum \$ zipWith (\u v -> a010051 u * a010051 v) [n+1..] \$ reverse [1..n-1] -- Reinhard Zumkeller, Nov 10 2012, Oct 19 2011 (PARI) a(n)=my(s); forprime(p=2, n-1, s+=isprime(2*n-p)); s \\ Charles R Greathouse IV, Mar 08 2013 CROSSREFS Cf. A071681 (subsequence for prime n only). Cf. A092953. Bisection of A117929 (even part). - Alois P. Heinz, Nov 14 2012 Sequence in context: A047931 A258571 A033618 * A270966 A138139 A259578 Adjacent sequences: A061354 A061355 A061356 * A061358 A061359 A061360 KEYWORD nonn,easy AUTHOR Amarnath Murthy, Apr 28 2001 EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001 STATUS approved

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Last modified November 29 07:51 EST 2023. Contains 367429 sequences. (Running on oeis4.)