A268035: a(n) is the smallest integer m such that each of the numbers m, 2*m, 3*m, ..., n*m is the sum of two successive primes.

a268305.py

"""
Author: David Radcliffe (2018-02-02)

I am interested in numbers that are sums of two successive primes:

   5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, ...

Note that 2+3=5, 3+5=8, 5+7=12, 7+11=18, and so on.
This is sequence A001043 in the On-Line Encyclopedia of Integer Sequences.

Somebody (I forget who) observed that if N is the sum of two successive primes,
then 2N is likely to be the sum of two successive primes as well. For example:

      5 +  7 = 12 and 11 + 13 = 24
     13 + 17 = 30 and 29 + 31 = 60
     19 + 23 = 42 and 41 + 43 = 84

This led me to wonder if there exist arbitrarily long sequences

   N, 2N, 3N, ..., kN

where each term is the sum of successive primes. So I proposed a new 
sequence which is defined as follows:

   a(k) = N if N is the smallest positive integer so that
   every number in the set {N, 2N, 3N, ..., kN} is the sum
   of two successive primes.

This is sequence A268035 in the On-Line Encyclopedia of Integer Sequences.
(See https://oeis.org/A268035)
"""

# https://github.com/hickford/primesieve-python
import primesieve 
from time import time

def a001043():
    """Generates numbers that are the sum of two consecutive primes."""
    it = primesieve.Iterator()
    prime = it.next_prime()
    while True:
        next_prime = it.next_prime()
        yield prime + next_prime
        prime = next_prime

def multi(k):
    """Generates numbers N so that each of the first k multiples of N
    is the sum of two consecutive primes."""
    seq = a001043()
    if k == 1:
        for s in seq: # This is an infinite loop! Use with caution.
            yield (s, s)
    current = next(seq)
    for first, last in multi(k-1): # This is also an infinite loop.
        last += first
        while current < last:
            current = next(seq)
        if current == last:
            yield (first, last)

def a268035(N):
    """Calculate and display the first N terms of Sequence A268035."""
    for n in range(1, N + 1):
        start_time = time()
        # We use the next() function to return the first result from an infinite iterator.
        print(n, next(multi(n))[0], time() - start_time)

a268035(10)

N.B. It is not difficult to prove that there does not exist an infinite sequence {N, 2N, 3N, 4N, ...} such that every term is the sum of two successive primes.