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

2,1

Comments

After a prime is reached it repeats itself infinitely. That's why this prime is then called the "home prime": it is the end of the calculation chain for a specific number.

Links

Table of n, a(n) for n=2..63.

Example

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.

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

Prog

(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

Crossrefs

Cf. A037274 (home primes).

Cf. A340393.

Sequence in context: A262189 A077336 A242174 * A288519 A240588 A013646

Adjacent sequences: A340391 A340392 A340393 * A340395 A340396 A340397

Keyword

nonn,base

Author

S. Brunner, Jan 06 2021

Status

approved