A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.
1, 1, 1, 2, 2, 3, 2, 2, 4, 3, 3, 5, 3, 3, 6, 5, 2, 6, 5, 4, 8, 4, 4, 7, 6, 5, 8, 7, 6, 12, 5, 3, 9, 5, 7, 11, 5, 4, 11, 8, 5, 13, 6, 7, 14, 8, 5, 11, 9, 8, 14, 7, 6, 13, 9, 7, 12, 7, 9, 18, 9, 6, 16, 8, 10, 16, 9, 7, 16, 14, 8, 17, 8, 8, 21, 10, 8, 17, 10, 11
Offset: 1
Examples
n=8: there are 16 pairs (i,j) with (i+j)/2=n*2=16; only two of them, (3,29) and (13,19), consist of primes, therefore a(8)=2.
Links
Crossrefs
Programs
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Haskell
a069360 n = sum [a010051' (4*n-p) | p <- takeWhile (<= 2*n) a000040_list] -- Reinhard Zumkeller, May 08 2014, Apr 09 2012
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Mathematica
Table[Length[Select[Range[0, 2*n], PrimeQ[2n-#] && PrimeQ[2n+#] &]], {n, 50}] (* Stefan Steinerberger, Nov 30 2007 *) Table[Boole[n == 1] + Sum[Boole[PrimeQ@ i] Boole[PrimeQ[4 n - i]] Mod[Sign[4 n - i], 4], {i, 3, 2 n}], {n, 80}] (* Michael De Vlieger, Dec 21 2016 *) Table[Count[IntegerPartitions[4n,{2}],?(AllTrue[#,PrimeQ]&)],{n,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Mar 09 2018 *) -
PARI
a(n)=my(s);forprime(p=2,2*n,s+=isprime(4*n-p));s \\ Charles R Greathouse IV, Apr 09 2012
Formula
For n > 1: a(n) = #{k | 2*n-k and 2*n+k are prime, 1<=k<=2*n}.
a(n) = Sum_{i=3..2n} isprime(i) * isprime(4n-i) * (sign(4n-i) mod 4), n > 1. - Wesley Ivan Hurt, Dec 18 2016
Extensions
Edited by Klaus Brockhaus, Nov 20 2007
a(1)=1, thanks to Charles R Greathouse IV, who noticed this; b-file adjusted.
Comments