I have tested all the primes up to 50,000,000 and did not find a single prime which satisfies the condition "sum of digits of prime number written in base7 divides by 3". E.g.
- 13 (Base10) = 16 (Base7) --> 7 (sum of digits in base 7)
- 1021 (Base10) = 2656 (Base7) --> 19
- 823541 (Base10) = 6666665 (Base7) --> 41
- 46941953 (Base10) = 1110000002 (Base7) --> 5
Here you can see the distribution of sums in base 7:
- COUNT(*) - the number of occurrences
- SUM7 - sum of digits in base7
- MIN(PRIME) - minimal prime in base10
- MAX(PRIME) - maximal prime in base10
As you can see sum7 of 9, 15, 21, 27, 33 are missing in the list, though other valid sums are widely represented. By 'valid sum' I mean that sum must be odd, because of "In an odd base, a number is odd if and only if it has an odd number of odd digits."
So what is the least prime whose sum of digits written in base7 divide by 3? Or is it possible to prove that all primes have such a feature?