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A366343
Numbers k such that every prime power q with q-1 | k-1 is a prime number.
0
2, 3, 5, 6, 11, 12, 14, 18, 20, 21, 23, 24, 26, 30, 35, 38, 39, 42, 44, 45, 47, 48, 51, 54, 56, 59, 60, 62, 66, 68, 69, 72, 74, 75, 77, 80, 83, 84, 86, 87, 90, 93, 95, 96, 98, 101, 102, 104, 107, 108, 110, 111, 114, 116, 117, 119, 122, 123, 126, 132, 135, 138, 140, 143, 144, 146, 147, 149, 150
OFFSET
1,1
COMMENTS
A number k has this property iff every field satisfying x^k = x is a prime field.
Equivalently, every ring satisfying x^k = x is additively generated by idempotents.
These numbers have been called "simple" (see MathOverflow link).
The natural density of the set appears to be approximately 0.462118.
LINKS
PROG
(SageMath)
def n_powers(n):
"""Computes the prime powers q with q-1 | n-1"""
return [x+1 for x in divisors(n-1) if (x+1).is_prime_power()]
def is_simple(n):
"""Checks if n is a simple number"""
return n > 1 and all(q.is_prime() for q in n_powers(n))
(PARI) isok(k) = if (k>1, fordiv(k-1, d, if (isprimepower(d+1) && !isprime(d+1), return(0)); ); return(1)); \\ Michel Marcus, Oct 07 2023
CROSSREFS
Sequence in context: A326612 A240949 A033159 * A199366 A332275 A318689
KEYWORD
nonn
AUTHOR
Martin Brandenburg, Oct 07 2023
STATUS
approved