In my recent post (October 24th 2023) titled Prime Digits, I mentioned primes whose digits are all prime. These primes constitute OEIS A019546 but 27253, while a member of this sequence, has an additional property that qualifies it for membership in OEIS A062088:
A062088 | Primes with every digit a prime and the sum of the digits a prime. |
The initial members of this sequence are:
2, 3, 5, 7, 23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3527, 3727, 5233, 5237, 5273, 5323, 5527, 7237, 7253, 7523, 7723, 7727, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257
While this makes 27253 special enough, the number also has the property that the products of its digits plus 1 and the product of its digits minus 1 are also prime. This combination of properties (prime, all digits are prime, sum of digits is prime, product of digits -1 is prime and product of digits +1 is prime) is rare indeed and up to 100,000 only the following numbers qualify: 23, 223, 22573, 25237, 27253, 32233, 32257, 32323, 33223, 35227, 52237, 57223, 72253, 75223. They are not listed in the OEIS. Table 1 shows the details.
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Table 1: permalink |
It can be seen that the majority of these numbers are permutations of the digits of 27253 and thus have the same sum of digits and product of digits \( \pm \) 1. Unfortunately, I failed to notice 25237 when it occurred and it will be a long time in terms of my diurnal age before I encounter the next one (32233). So I felt I should make a note of 27253 in this post.
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