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Search: id:A061357
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| A061357 |
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Number of 1<=k<n such that n-k and n+k are both primes. |
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7
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| 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; listen)
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OFFSET
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1,8
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COMMENT
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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 (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.)
(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)).
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]
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
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LINKS
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P. CAMI, Table of n, a(n) for n = 4..60000
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FORMULA
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a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007
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EXAMPLE
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a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.
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MAPLE
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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]
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CROSSREFS
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Cf. A071681 (subsequence for prime n only).
Sequence in context: A089993 A047931 A033618 this_sequence A138139 A127992 A067595
Adjacent sequences: A061354 A061355 A061356 this_sequence A061358 A061359 A061360
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KEYWORD
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nonn,easy
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 28 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001
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