A304875 -id:A304875 - cp's OEIS frontend

A304874 Greatest prime p1 < p2 such that n^2 = (p1 + p2)/2 and p2 is prime.

3, 7, 13, 19, 31, 37, 61, 79, 97, 103, 139, 157, 193, 223, 241, 271, 317, 349, 379, 439, 421, 487, 521, 619, 661, 719, 757, 829, 881, 883, 1009, 1087, 1063, 1213, 1291, 1291, 1429, 1511, 1579, 1669, 1741, 1831, 1879

Offset: 2

Hugo Pfoertner, May 20 2018

nonn

Each square > 1 can be written as the average of 2 primes p1 < p2. a(n) gives the greatest prime p1 such that n^2 = (p1 + p2) / 2. The corresponding p2 is provided in A304875.

			a(2) = 3 because 2^2 = 4 = (3 + 5)/2,
a(7) = 37 because 7^2 = 49 = (37 + 61)/2 and none of the primes p1 = 41, 43 or 47 leads to a prime p2.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A172989, A304875.

a(n) = n^2 - A172989(n) = A304875(n) - 2*A172989(n).

A304903 Least prime p such that 2*n^2 - p is prime.

Original entry on oeis.org

3, 5, 3, 3, 5, 19, 19, 5, 3, 3, 5, 7, 3, 7, 3, 7, 5, 3, 3, 5, 31, 7, 23, 13, 31, 5, 19, 13, 11, 43, 19, 17, 3, 3, 13, 7, 31, 5, 13, 3, 11, 7, 19, 23, 3, 61, 5, 3, 7, 5, 61, 37, 5, 3, 3, 7, 19, 3, 7, 31, 7, 5, 13, 3, 5

Offset: 2

Hugo Pfoertner, May 20 2018

nonn

Each square > 1 can be written as the average of 2 primes p1 < p2. a(n) gives the least prime p1 such that n^2 = (p1 + p2) / 2. The corresponding p2 is provided in A304904.

			a(5) = 3 because 2*5^2 - 3 = 47 is prime,
a(7) = 19 because 2*7^2 - 19 = 71 is prime, whereas all of 98 - 3 = 95, 98 - 5 = 93, 98 - 7 = 91, 98 - 11 = 87, 98 - 13 = 85 and 98 - 17 = 81 are composite.

Cf. A304874, A304875, A304904, A304905.

f[n_] := Block[{p = 2}, While[ !PrimeQ[2 n^2 - p], p = NextPrime@ p]; p]; Array[f, 65, 2] (* Robert G. Wilson v, May 20 2018 *)

a(n) = forprime(p=3, , if(ispseudoprime(2*n^2-p), return(p))) \\ Felix Fröhlich, May 20 2018

a(n) = n^2 - A304905(n) = A304904(n) - 2*A304905(n).

A304904 Greatest prime p such that 2*n^2 - p is prime.

Original entry on oeis.org

5, 13, 29, 47, 67, 79, 109, 157, 197, 239, 283, 331, 389, 443, 509, 571, 643, 719, 797, 877, 937, 1051, 1129, 1237, 1321, 1453, 1549, 1669, 1789, 1879, 2029, 2161, 2309, 2447, 2579, 2731, 2857, 3037, 3187, 3359, 3517

Offset: 2

Hugo Pfoertner, May 20 2018

nonn

Each square > 1 can be written as the average of 2 primes p1 < p2. a(n) gives the greatest prime p2 such that n^2 = (p1 + p2) / 2. The corresponding p1 is provided in A304903.

			a(6) = 67 because 2*6^2 - 67 = 5 is prime whereas 72 - 71 = 1 is not a prime.

Cf. A304874, A304875, A304903, A304905.

a304903(n) = forprime(p=3, , if(ispseudoprime(2*n^2-p), return(p)))
a(n) = 2*n^2-a304903(n) \\ Felix Fröhlich, May 20 2018

a(n) = n^2 + A304905(n) = A304903(n) + 2*A304905(n).

A305125 Number of ways to express n^2 as average of two primes p1 < p2.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 3, 3, 10, 8, 8, 17, 9, 11, 27, 11, 12, 27, 14, 21, 39, 17, 19, 36, 28, 22, 48, 25, 24, 75, 30, 25, 68, 35, 56, 68, 37, 40, 93, 54, 43, 103, 42, 52, 125, 51, 49, 117, 64, 76, 130, 63, 56, 135, 99, 78, 151, 76, 73, 198

Offset: 1

Hugo Pfoertner, May 26 2018

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A002375, A172989, A304874, A304875, A304903, A304904, A304905, A305126.

f[n_]:=Length[Select[2 n - Prime[Range[2, PrimePi[n]]], PrimeQ]]; Table[f[n^2], {n, 60}] (* Vincenzo Librandi, May 26 2018 *)

for (n=1,60,k=0;n2=2*n*n;forprime(p=3,n2/2,if(isprime(n2-p),k++));print1(k,", "))

a(n) = A002375(n^2).

cp's OEIS Frontend

A304874 Greatest prime p1 < p2 such that n^2 = (p1 + p2)/2 and p2 is prime.

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A304903 Least prime p such that 2*n^2 - p is prime.

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A304904 Greatest prime p such that 2*n^2 - p is prime.

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A305125 Number of ways to express n^2 as average of two primes p1 < p2.

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