

A069360


Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.


7



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
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OFFSET

1,4


COMMENTS

The Goldbach conjecture, if true, would imply a(n) > 0.
Row lengths of table A260689, n > 1.  Reinhard Zumkeller, Nov 17 2015


LINKS

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture


FORMULA

For n > 1: a(n) = #{k  2*nk and 2*n+k are prime, 1<=k<=2*n}.
a(n) = Sum_{i=3..2n} isprime(i) * isprime(4ni) * (sign(4ni) mod 4), n > 1.  Wesley Ivan Hurt, Dec 18 2016


EXAMPLE

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.


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 *)


PROG

(Haskell)
a069360 n = sum [a010051' (4*np)  p < takeWhile (<= 2*n) a000040_list]
 Reinhard Zumkeller, May 08 2014, Apr 09 2012
(PARI) a(n)=my(s); forprime(p=2, 2*n, s+=isprime(4*np)); s \\ Charles R Greathouse IV, Apr 09 2012


CROSSREFS

Bisection of A002375.
Cf. A082467 (least k such that nk and n+k are both primes), A134677 (records), A134678 (where records occur), A135146 (index of first occurrence of n).
Cf. A000040, A010051, A260689.
Sequence in context: A230296 A278317 A086454 * A175509 A213023 A068050
Adjacent sequences: A069357 A069358 A069359 * A069361 A069362 A069363


KEYWORD

nonn,easy,nice


AUTHOR

Reinhard Zumkeller, Apr 15 2002


EXTENSIONS

Edited by Klaus Brockhaus, Nov 20 2007
a(1)=1, thanks to Charles R Greathouse IV, who noticed this; bfile adjusted.


STATUS

approved



