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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.
2
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
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
Sequence in context: A327306 A053838 A342651 * A117167 A117169 A274263
KEYWORD
nonn
AUTHOR
Bob Selcoe and Jamie Morken, Aug 07 2016
STATUS
approved