A275768 a(n) is the number of ways to express n = (prime(i) + prime(j))/2 when (prime(i) - prime(j))/2 also is prime.

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 1, 1, 0, 5, 0, 1, 0, 0, 0, 5, 0, 1, 0, 1, 0, 5, 0, 0, 1, 1, 0, 6, 0, 1, 1, 1, 0, 5, 0, 2, 0, 0, 0, 5, 0, 2, 0, 0, 0, 10, 0, 0, 0, 1, 0, 8, 0, 0, 1, 2, 0, 6, 0, 0, 0, 2, 0, 8, 0, 0, 1

Offset: 0

Author

Bob Selcoe and Jamie Morken, Aug 07 2016

Keywords

• nonn

Comments

It appears that peaks occur when n is a multiple of primorial(k), and the peaks amplify as k increases.

a(5) = 1 is the only term > 0 where odd n is not a multiple of 3. Proof: let prime C = (prime(i) - prime(j))/2 and D = (prime(i) + prime(j))/2. Then D is odd iff C=2. Odd D must be a multiple of 3 unless prime(j) is not a multiple of 3; thus D is not a multiple of 3 only when prime(j) = 3.

From Michael De Vlieger, Apr 30 2017: (Start)

First occurrence of values k of a(n) for 0 <= n <= 10^4, with -1 meaning value does not occur in range of n: {0, 5, 8, 18, -1, 24, 42, 96, 66, 198, 60, 126, 90, 150, 234, 408, 120, 294, 240, 378, 582, 270, ...}.

Does a(n) = 4 occur for any n?

Order of appearance of values k of a(n): {0, 1, 2, 3, 5, 6, 10, 8, 12, 7, 16, 11, 13, 9, 26, 14, 18, 21, 17, 31, 25, 19, 15, 38, 30, ...}.

a(A060735(n)) = {0, 0, 0, 0, 2, 3, 5, 5, 10, 12, 16, 13, 16, 26, 38, 54, 59, 64, 74, 79, 87, 89, 98, 124, ...}.

a(A002110(n)) = {0, 0, 0, 5, 26, 124, 852, 7550, 86125, ...}. (End)

Number of Goldbach partitions (p,q) of 2n such that |q-p|/2 is prime. For example, a(8) = 2; 2*8 = 16 has 2 Goldbach partitions (3,13) and (5,11). Both |13-3|/2 = 5 and |11-5|/2 = 3 are prime, so a(8) = 2. - Wesley Ivan Hurt, Apr 03 2018

Examples

			a(8) = 2 because (13-3)/2 = 5 and (13+3)/2 = 8; and (11-5)/2 = 3 and (11+5)/2 = 8.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Jamie Morken, Graph showing formation of primorial bands for n = 0..100000
• Jamie Morken, Graph showing primorial peaks for n = 30000..30276

Crossrefs

Cf. A000040, A010051.

Programs

Mathematica

Table[Count[Map[{2 n - #, #} &, Range@ n], w_ /; And[Times @@ Boole@ Map[PrimeQ, w] == 1, PrimeQ[(Subtract @@ w)/2]]], {n, 0, 81}] (* Michael De Vlieger, Apr 30 2017 *)
(*Example of a program to find first 1000 terms of a(n)*)
For[z = 0, z < 1000, z++,
countOfPrimes = 0;
countOfPrimes2 = 0;
countOfPrimes3 = 0;
PnToUse = z;
distanceToCheck = PnToUse;
For[i = 0, i < distanceToCheck, i++,
   If[PrimeQ[2*PnToUse - i],
    countOfPrimes++ If[PrimeQ[(2*PnToUse - i) - PnToUse],
      countOfPrimes2++ If[PrimeQ[i], countOfPrimes3++]],]]
  Print[countOfPrimes3]]
(* Jamie Morken, May 20 2017 *)

Formula

a(n) = Sum_{i=1..n} A010051(n-i) * A010051(2n-i) * A010051(i). - Wesley Ivan Hurt, Apr 03 2018