A061357 Number of 0<k<n such that n-k and n+k are both primes.

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

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 0<b<n-1 and p,q are primes. - Erin Noel and George Panos (erin.m.noel(AT)rice.edu), Jun 27 2006

a(n) = sum (A010051(2*n - p): p prime < n). [Reinhard Zumkeller, Oct 19 2011]

a(n) is also the number of partitions of 2*n into two distinct primes. See the first formula by T. D. Noe, and the Alois P. Heinz, Nov 14 2012, crossreference. - Wolfdieter Lang, May 13 2016

All 0<k<n are coprime to n. - Jamie Morken, Jun 02 2017

a(n) is the number of appearances of n in A143836. - Ya-Ping Lu, Mar 05 2023

Links

Pierre CAMI, Table of n, a(n) for n = 1..60000

Formula

a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007

a(n) = sum(A010051(n-k)*A010051(n+k): 1 <= k < n). - Reinhard Zumkeller, Nov 10 2012

a(n) = sum_{i=2..n-1} A010051(i)*A010051(2n-i). [Wesley Ivan Hurt, Aug 18 2013]

Example

a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.

Maple

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); # 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