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 ± 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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