A316941 The natural numbers sequence where every composite is replaced by a prime, according to the rule explained in the Comments section (a(n) = -1 if no such prime is reached).

1, 2, 3, 211, 5, 23, 7, 2692213, 311, 773, 11, 3251, 13, 313, 1129, 3313788967, 17, 29, 19, 724553, 37, 211, 23, 2692213, 773, 3251, 313, 3313788967, 29, 3181, 31, 210527, 311, 3581, 1129, 7529, 37, 373, 313, 232357, 41, 19181, 43, 2111, 3119014487, 223, 47, 310345345771837, 31079, 3197071252784831

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jul 17 2018

Keywords

• base
• nonn

Comments

Only the composites are replaced by a prime; 1 and the primes themselves stay as they are. A composite "c" is replaced by the concatenation [az], where "a" is the smallest factor > 1 of "c" and "z" the biggest factor < "c". This transforms for instance 28 in [214] (because 2 x 14 is 28) and not in [128] (1 x 28) or [47] (4 x 7). The procedure is iterated from there until the concatenation produces a prime [214 becomes 2107, then 7301, 71043, 323681, 431751, 3143917, 7449131, 12895779, 34298593, 74899799, 135761523, 345253841, 941366901 which ends into the prime 3313788967, prime that will thus be a(28)].

Examples

			As the first three natural numbers (1, 2 and 3) are not composites, they stay as they are. Then 4 becomes 22, and 22 produces the prime 211; 5 is a prime; 6 becomes the prime 23; 7 is a prime; 8 ends on the prime 2692213; 9 becomes 33 and 33 produces the prime 311; 10 produces the chain 25, 55, 511, 773 (prime); etc.
Some terms of this sequence are huge: a(158) is a 70-digit prime, for instance.
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..175

Crossrefs

Cf. A002808 (the composite numbers).

Programs

Python

from sympy import factorint
def A316941(n):
    while n>1 and sum((f:=factorint(n)).values()) > 1:
        n = int(str(p:=min(f))+str(n//p))
    return n # Chai Wah Wu, Aug 02 2024