A340394 Base-independent home primes: the prime that is finally reached when you treat the prime factors of n in ascending order as digits of a number in base "greatest prime factor + 1" and repeat this until a prime is reached (a(n) = -1 if no prime is ever reached).
2, 3, 41, 5, 11, 7, 41, 23, 17, 11, 43, 13, 23, 23, 3407, 17, 47, 19, 89, 31, 47, 23, 1279, 47, 41, 223, 151, 29, 167, 31, 431, 47, 53, 47, 367, 37, 59, 71, 521, 41, 263, 43, 359, 131, 71, 47, 683, 223, 107, 71, 433, 53, 191, 71, 11807, 79, 89, 59, 3023, 61, 167, 223
Offset: 2
Examples
For n=4 we get the base-independent home prime 41 through this chain of calculations: 4 = 2 * 2 -> 22_3 (base 3 because 3 = greatest prime factor (2) + 1) 22_3 = 8_10 = 2 * 2 * 2 -> 222_3 222_3 = 26_10 = 2 * 13 -> 2D_14 2D_14 = 41_10, which is a prime. This gives us 41 as our home prime for n = 4, 8, 26 and 41.
Programs
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Maple
b:= n-> (l-> (m-> add(l[-i]*m^(i-1), i=1..nops(l)))(1+ max(l)))(map(i-> i[1]$i[2], sort(ifactors(n)[2]))): a:= n-> `if`(isprime(n), n, a(b(n))): seq(a(n), n=2..77); # Alois P. Heinz, Jan 09 2021 -
PARI
f(n) = my(f=factor(n), list=List()); for (k=1, #f~, for (j=1, f[k, 2], listput(list, f[k, 1]))); fromdigits(Vec(list), vecmax(f[, 1])+1); \\ A340393 a(n) = my(p); while (! isprime(p = f(n)), n = p); p; \\ Michel Marcus, Jan 07 2021
Comments