A330210 Numbers that can be expressed as the sum of 2 prime numbers in a prime number of different ways.

10, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 38, 40, 44, 48, 52, 54, 56, 62, 64, 68, 70, 74, 76, 78, 82, 86, 94, 96, 98, 104, 112, 124, 128, 130, 136, 140, 144, 148, 156, 158, 164, 168, 174, 176, 178, 186, 188, 192, 194, 198, 206, 208, 210, 216, 218, 222, 224

Offset: 1

Author

Pietro Saia, Dec 05 2019

Keywords

• nonn

Examples

			24 can be expressed as the sum of 2 prime numbers in 3 different ways (5+19, 7+17, and 11+13), and 3 is prime.
		

Crossrefs

Cf. A000040, A014091, A061358.

Programs

Mathematica

Select[Range[2, 224, 2], PrimeQ@ Length@ IntegerPartitions[#, {2}, Prime@ Range@ PrimePi@ #] &] (* Giovanni Resta, Dec 06 2019 *)

Python

import math
from sympy import isprime
def main(n):
    x = {}
    a = 1
    b = 1
    for i in range(2, n):
        x[i] = []
        while a < i:
            if a + b == i:
                x[i].append(str(a) + "+" + str(b))
            b += 1
            if b == i:
                a += 1
                b = 1
        a = 1
        b = 1
    for i in x:
        x[i] = x[i][0:math.ceil(len(x[i])/2)]
    x[2] = ["1+1"]
    newdict = {}
    for i in x:
        newdict[i] = []
        for j in x[i]:
            if isprime(int(j.split("+")[0])) and isprime(int(j.split("+")[1])):
                newdict[i].append(j)
    finaloutput = []
    for i in newdict:
        if isprime(len(newdict[i])):
            finaloutput.append(i)
    return finaloutput
def a(n):
    x = 0
    while len(main(x)) != n:
        x += 1
    return main(x)[-1]