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(Python)
import sys
import math
from sympy.ntheory import primefactors
from sympy.ntheory import primerange
def intersection(lst1, lst2):
lst3 = [value for value in lst1 if value in lst2]
return len(lst3)
n_primes=1000000
factors=[primefactors(n) for n in range(0, n_primes)]
primes=list(primerange(0, n_primes))
sequence=[4]
sums=[sequence[0]]
prime_factors=[f for f in factors[sequence[0]]]
big_n=8
while len(sequence)<big_n:
new_a=False
a=sequence[-1]+1
while intersection(factors[a], prime_factors)!=0:
a+=1
n=len(sequence)
while not new_a:
new_sum=[a+sum for sum in sums]
prime_sum=False
for sum in new_sum:
if sum in primes:
prime_sum=True
if not prime_sum and a not in primes:
sequence.append(a)
print(a, end=", ")
sys.stdout.flush()
sums=sums+new_sum+[a]
sums = list(dict.fromkeys(sums))
prime_factors=prime_factors+factors[a]
new_a=True
else:
a+=1
while a in primes or intersection(factors[a], prime_factors)!=0:
a+=1
print()
(Python)
from math import gcd
from sympy import isprime
from itertools import islice
def agen(start=4): # generator of terms
alst, k, sums = [start], start+1, {0} | {start}
while True:
yield alst[-1]
while any(gcd(k, an) != 1 for an in alst) or \
any(k+s not in sums and isprime(k+s) for s in sums):
k += 1
alst.append(k)
sums.update([k + s for s in sums])
k += 1
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