cp's OEIS Frontend

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.

%I A275768 #47 Apr 08 2018 09:21:21
%S A275768 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,
%T A275768 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,
%U A275768 0,1,2,0,6,0,0,0,2,0,8,0,0,1
%N 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.
%C A275768 It appears that peaks occur when n is a multiple of primorial(k), and the peaks amplify as k increases.
%C A275768 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.
%C A275768 From _Michael De Vlieger_, Apr 30 2017: (Start)
%C A275768 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, ...}.
%C A275768 Does a(n) = 4 occur for any n?
%C A275768 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, ...}.
%C A275768 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, ...}.
%C A275768 a(A002110(n)) = {0, 0, 0, 5, 26, 124, 852, 7550, 86125, ...}. (End)
%C A275768 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
%H A275768 Michael De Vlieger, <a href="/A275768/b275768.txt">Table of n, a(n) for n = 0..10000</a>
%H A275768 Jamie Morken, <a href="http://imgur.com/gcw7S39">Graph showing formation of primorial bands for n = 0..100000</a>
%H A275768 Jamie Morken, <a href="http://imgur.com/x7yDW1e">Graph showing primorial peaks for n = 30000..30276</a>
%F A275768 a(n) = Sum_{i=1..n} A010051(n-i) * A010051(2n-i) * A010051(i). - _Wesley Ivan Hurt_, Apr 03 2018
%e A275768 a(8) = 2 because (13-3)/2 = 5 and (13+3)/2 = 8; and (11-5)/2 = 3 and (11+5)/2 = 8.
%t A275768 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 *)
%t A275768 (*Example of a program to find first 1000 terms of a(n)*)
%t A275768 For[z = 0, z < 1000, z++,
%t A275768 countOfPrimes = 0;
%t A275768 countOfPrimes2 = 0;
%t A275768 countOfPrimes3 = 0;
%t A275768 PnToUse = z;
%t A275768 distanceToCheck = PnToUse;
%t A275768 For[i = 0, i < distanceToCheck, i++,
%t A275768    If[PrimeQ[2*PnToUse - i],
%t A275768     countOfPrimes++ If[PrimeQ[(2*PnToUse - i) - PnToUse],
%t A275768       countOfPrimes2++ If[PrimeQ[i], countOfPrimes3++]],]]
%t A275768   Print[countOfPrimes3]]
%t A275768 (* _Jamie Morken_, May 20 2017 *)
%Y A275768 Cf. A000040, A010051.
%K A275768 nonn
%O A275768 0,9
%A A275768 _Bob Selcoe_ and _Jamie Morken_, Aug 07 2016