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Home prime
The home prime of n, denoted by HP(n) is found by concatenating the prime factors of n, repeatedly, until a prime is reached. The notion of a home prime depends on the base (except in the case where n itself is prime).
Notations for other bases of 10 are HPbase(n) for base b.
While it is expected that every n in every base has a home prime, experimental evidence indicates that these chains can get quite long.
Contents
Base 10
For example, the home prime of 25 is 773 because
- 25 = 5 × 5
- 55 = 5 × 11
- 511 = 7 × 73
and finally 773 is prime.
The length of such chains is also of interest, in this case #HP(25) = 3.
Base 2
HP2(10):
- 1010 = 2 × 5 (10 × 101)
- 2110 = 3 × 7 (11 × 111)
- 31 prime
