A384306 Primes whose sum of digits in both base 8 and base 10 are recursively prime down to 2, 3, 5, or 7.
2, 3, 5, 7, 131, 311, 887, 1013, 1949, 2399, 2621, 2957, 3251, 3323, 3701, 4289, 4919, 4973, 5099, 5101, 5477, 5927, 5981, 6359, 6599, 6779, 6863, 8069, 8447, 8573, 8627, 8669, 8951, 9677, 10141, 10181, 10211, 10589, 10631, 11399, 11597, 12101, 12479, 12659, 12983
Offset: 1
Examples
a(5) = 131:
In base 8: 131 = 203_8 -> 2+0+3 = 5 -> 5 is prime -> ends in 5.
In base 10: 1+3+1 = 5 -> 5 is prime -> ends in 5.
All intermediate values for both bases are primes, and the last values are in {2,3,5,7}.
a(6) = 887:
In base 8: 887 = 1567_8 -> 1+5+6+7 = 19 -> 19 is prime -> 19 = 23_8 -> 2+3 = 5-> 5 is prime -> ends in 5.
In base 10: 8+8+7 = 23 -> 23 is prime -> 2+3 = 5 -> 5 is prime -> ends in 5.
All intermediate values for both bases are primes, and the last values are in {2,3,5,7}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..11089 (first 1635 terms from Jean-Louis Lascoux)
Programs
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Maple
q:= (n, k)-> isprime(n) and (n
q(x, 8) and q(x, 10), [$1..20000])[]; # Alois P. Heinz, May 27 2025 -
PARI
isokb(p,b) = while(1, my(s=sumdigits(p, b)); if (! isprime(s), return(0)); if (s
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Python
from gmpy2 import digits, is_prime def rp(n, b): return is_prime(n) and (n < b or rp(sum(map(int, digits(n, b))), b)) def ok(n): return rp(n, 10) and rp(n, 8) print([k for k in range(2, 13000) if ok(k)]) # Michael S. Branicky, May 27 2025
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