1 is a product of 0 primes such that every 2 positive integers that multiply to it add up to a prime.

1*1=1, 1+1=2

2 is a product of 1 prime such that every 2 positive integers that multiply to it add up to a prime.

1*2=2, 1+2=3

6=2*3 is a product of 2 primes such that every 2 positive integers that multiply to it add up to a prime.

1*6=6, 1+6=7
2*3=6, 2+3=5

30=2*3*5 is a product of 3 primes such that every 2 positive integers that multiply to it add up to a prime.

1*30=30, 1+30=31
2*15=30, 2+15=17
3*10=30, 3+10=13
5*6=30, 5+6=11

1. What is the smallest product of 4 primes such that every 2 positive integers that multiply to it add up to a prime?
2. What is the smallest product of 5 primes such that every 2 positive integers that multiply to it add up to a prime?

Except for '1', only even numbers can test positive because (1,odd) would be one pair of factors, and 1 + odd is even, and 2 is the only even prime.  I searched the even numbers up to 10,000,000

n  smallest
1   2
2   6
3   30
4   210
5   186162
Which is oeis A293756 with no other entries, which I discovered after the fact.

Here are the other integers that met the criteria for 4 and 5.

4: [210, 330, 462, 1870, 4218, 5590, 6042, 7638, 13962, 14410, 22570, 24682, 26202, 32838, 35098, 39678, 43498, 43990, 52782, 62458, 62538, 93262, 125302, 127290, 130782, 152310, 155170, 201910, 221622, 227110, 236062, 272902, 296718, 300490, 309358, 331710, 339262, 355762, 410278, 418342, 476578, 489742, 670558, 685518, 691062, 710682, 810418, 867262, 868330, 950482, 989422, 1011558, 1019530, 1031298, 1172938, 1184682, 1274938, 1307590, 1336602, 1489818, 1500618, 1558342, 1619662, 1685170, 1804870, 1908082, 1925610, 1940422, 2043490, 2066758, 2263438, 2301258, 2325810, 2401870, 2462122, 2584558, 2785762, 2896738, 2998318, 3044710, 3109390, 3117678, 3244858, 3347458, 3395478, 3452178, 3583918, 3746338, 4202958, 4206610, 4274418, 4346818, 4423182, 4758502, 4978030, 5263630, 5283982, 5353342, 5435362, 5703478, 5721802, 5735262, 5880322, 5968438, 5988442, 6046870, 6445938, 6458782, 6870910, 7253878, 7283698, 7315378, 7713178, 8586622, 8870178, 8962798, 9052738, 9195778, 9838378],
5: [186162, 899970, 3047410]

---------------
def prime_factor(n):
    """  for integer n, return a list of all the prime factors """
    top = n // 2
    factors = []
    
    for i in range(2,top+1):
        while n/i % 1 == 0:
            factors.append(int(i))
            n = n/i
            if n == 1:
                return factors
    if n != 1:
        factors.append(int(n))
    return factors

def isprime(n):
    '''check if integer n is a prime'''
    n = abs(int(n))
    if n < 2:
        return False
    if n == 2:
        return True    
    if not n & 1:
        return False
    for x in range(3, int(n**0.5)+1, 2):
        if n % x == 0:
            return False
    return True

def factorpairs(n):
    """  for integer n, return a list of all the factors including 1 and n """
    # ans = [1,n]
    ans = []
    for i in range(2,2+int(n**.5)):
        if n%i == 0:
            ans.append(i)
            ans.append(int(n/i))
    ans = sorted(ans)
    finalans = [[1,n]]
    while len(ans)  >  1:
        finalans.append([ans[0],ans[-1]])
        ans.pop()
        ans.pop(0)
    if len(ans) == 1:
        finalans.append([ans[0],ans[-1]])
        
    return finalans

def test(n):
    for t in factorpairs(n):
        if not isprime(sum(t)):
            return False
    return True

big = 10000000
goods = []
goodDict = {}

for i in range(0,big,2):
    if test(i):
        goods.append(i)
        size = len(prime_factor(i))
        if size not in goodDict:
            goodDict[size] = [i]
        else:
            goodDict[size].append(i)

primes = [i for i in range(10000) if isprime(i)]
print(goodDict)

Posted by Larry on 2023-04-19 08:13:05