Special Primes

I'm surprised that I haven't written about these sorts of primes before. The primes form OEIS A092529:

A092529  primes $p$ such that both the digit sum of $p$ plus $p$ and the digit product of $p$ plus $p$ are also primes (zeroes are not permitted).

Up to 40000, there are 136 such primes and they are (permalink):

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1867, 2273, 2543, 2633, 3449, 4211, 4217, 4273, 4547, 4729, 5861, 6121, 6529, 6637, 6653, 6761, 6857, 6949, 7681, 8273, 8431, 8837, 8839, 9649, 9689, 11251, 11657, 11677, 11897, 12379, 12553, 13163, 13457, 13523, 13697, 13729, 13877, 13879, 14423, 14533, 14537, 14957, 15121, 15217, 15277, 15361, 15413, 15451, 15619, 15727, 15859, 16319, 16427, 16993, 17183, 17299, 17837, 18229, 18287, 18517, 19381, 19457, 19583, 21379, 21467, 21577, 21737, 21751, 21977, 21991, 22123, 22259, 22369, 22549, 22921, 23117, 23269, 23599, 23719, 24499, 24527, 25111, 25153, 25577, 25771, 25847, 25913, 25997, 26251, 26699, 26927, 27427, 28181, 29153, 29179, 32173, 32687, 32957, 32971, 33413, 33547, 33581, 33587, 33769, 33851, 34313, 34667, 35251, 35257, 35323, 35521, 35569, 35831, 36229, 36469, 36559, 36919, 37321, 37369, 37547, 37871, 38351, 38959, 39161, 39521

Let's just check the first number in this sequence, 163. The digit sum is 10 and the digit product is 18. Now 163 + 10 = 173 which is prime and 163 + 18 = 181 is also prime.

Interestingly if we consider subtraction instead of addition then no such primes exist. That is to say that, even up to one million, there are no primes such that $p$ minus the digit sum of $p$ and $p$ minus the digit product of $p$ are also primes (with zeroes not permitted). I'm not sure why this is so.

We can thin the ranks of the above primes if we require that the sum and product of the squares of the digits also form primes when added to the original prime. Only four numbers such satisfy all these criteria in the range up to 40000. These are 1423, 13697, 14533 and 33413 (permalink). Let's examine the first of these numbers 1423. The sum of digits is 10 and the product is 24. The digits squared are 1, 16, 4 and 9 with a sum of 30 and a product of 576. Thus we have:

• 1423 + 10 = 1433 (prime)
• 1423 + 24 = 1447 (prime)
• 1423 + 30 = 1453 (prime)
• 1423 + 576 = 1999 (prime)

If we extend the range up to one million, there are 48 numbers that satisfy:

1423, 13697, 14533, 33413, 53419, 57529, 61991, 71569, 91129, 125789, 128153, 132527, 132679, 143477, 149161, 159463, 223423, 238649, 275929, 284831, 288493, 297613, 316343, 337261, 343639, 367819, 375227, 441797, 447791, 498733, 512521, 573829, 574969, 582937, 613673, 626723, 722333, 723923, 728681, 735283, 746533, 748883, 752273, 762539, 766531, 836917, 869951, 872959

To thin this sequence further, let's impose additional criteria, specifically that the number plus the sum of the cubes of the digits and the number plus the product of the cubes of the digits must be prime as well. Here are the numbers up to one million: 125789, 132527, 573829 and 752273. Let's look at the first of these numbers, 125789:

• 125789 + 32 = 125821 (prime)
• 125789 + 5040 = 130829 (prime)
• 125789 + 224 = 126013 (prime)
• 125789 + 25401600 = 25527389 (prime)
• 125789 + 1718 = 127507 (prime)
• 125789 + 128024064000 = 128024189789 (prime)

Notice how the product of the squares and cubes of the digits are larger than the number itself. If we remove the requirement that the initial number be prime then a few more numbers satisfy the criteria for digits, digits squared and digits cubed. These are 13969, 65821, 125789, 132527, 349789, 537881, 545123, 573829 and 752273 (primes are shown in red).