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%I #12 Dec 07 2019 00:50:13
%S 10,14,16,18,20,22,24,26,28,30,32,38,40,44,48,52,54,56,62,64,68,70,74,
%T 76,78,82,86,94,96,98,104,112,124,128,130,136,140,144,148,156,158,164,
%U 168,174,176,178,186,188,192,194,198,206,208,210,216,218,222,224
%N Numbers that can be expressed as the sum of 2 prime numbers in a prime number of different ways.
%e 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.
%t Select[Range[2, 224, 2], PrimeQ@ Length@ IntegerPartitions[#, {2}, Prime@ Range@ PrimePi@ #] &] (* _Giovanni Resta_, Dec 06 2019 *)
%o (Python 3)
%o import math
%o from sympy import isprime
%o def main(n):
%o x = {}
%o a = 1
%o b = 1
%o for i in range(2, n):
%o x[i] = []
%o while a < i:
%o if a + b == i:
%o x[i].append(str(a) + "+" + str(b))
%o b += 1
%o if b == i:
%o a += 1
%o b = 1
%o a = 1
%o b = 1
%o for i in x:
%o x[i] = x[i][0:math.ceil(len(x[i])/2)]
%o x[2] = ["1+1"]
%o newdict = {}
%o for i in x:
%o newdict[i] = []
%o for j in x[i]:
%o if isprime(int(j.split("+")[0])) and isprime(int(j.split("+")[1])):
%o newdict[i].append(j)
%o finaloutput = []
%o for i in newdict:
%o if isprime(len(newdict[i])):
%o finaloutput.append(i)
%o return finaloutput
%o def a(n):
%o x = 0
%o while len(main(x)) != n:
%o x += 1
%o return main(x)[-1]
%Y Cf. A000040, A014091, A061358.
%K nonn
%O 1,1
%A _Pietro Saia_, Dec 05 2019
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