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

0,9

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

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A000040, A010051.

Sequence in context: A327306 A053838 A342651 * A117167 A117169 A274263

Adjacent sequences: A275765 A275766 A275767 * A275769 A275770 A275771

Keyword

nonn

Author

Bob Selcoe and Jamie Morken, Aug 07 2016

Status

approved