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A323214
Composite numbers k such that p^(k-1) == 1 (mod k) for every prime p strongly prime to k.
0
4, 6, 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721
OFFSET
1,1
COMMENTS
A positive number k <= n is strongly prime to n if and only if k is prime to n and k does not divide n-1. See A322937 and the link to 'Strong Coprimality'.
Apparently essentially the Carmichael numbers A002997.
LINKS
K. Bouallègue, O. Echi and R. G. E. Pinch, Korselt numbers and sets, International Journal of Number Theory, 6 (2010), 257-269.
A. Korselt, G. Tarry, I. Franel and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), 142-144.
Peter Luschny, Strong Coprimality
C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to 25*10^9, Math. Comp., 35 (1980), 1003-1026.
V. Šimerka, Zbytky z arithmetické posloupnosti, (On the remainders of an arithmetic progression), Časopis pro pěstování matematiky a fysiky. 14 (1885), 221-225.
L. Wang, The Korselt set of a power of a prime, International Journal of Number Theory, 14 (2018), 233-240.
EXAMPLE
2, 3 and 5 are not in this sequence because primes are not in this sequence.
4 and 6 are in this sequence because there are no primes strongly prime to 4 respectively 6.
For n = 1729 there are 1296 test cases using the definition of A002997 but only 264 test cases using the definition of a(n).
PROG
(Sage)
def is_strongCarmichael(n):
if n == 1 or is_prime(n): return False
for k in (1..n):
if is_prime(k) and not k.divides(n-1) and is_primeto(k, n):
if power_mod(k, n-1, n) != 1: return false
return true
def A323214_list(len):
return [n for n in (1..len) if is_strongCarmichael(n)]
print(A323214_list(600000))
(Julia)
using IntegerSequences
PrimesPrimeTo(n) = (p for p in Primes(n) if isPrimeTo(p, n))
function isStrongCarmichael(n)
if isComposite(n)
for k in PrimesPrimeTo(n)
if ! Divides(k, n-1)
if powermod(k, n-1, n) != 1
return false
end
end
end
return true
end
return false
end
L323214(len) = [n for n in 1:len if isStrongCarmichael(n)]
L323214(30000) |> println
CROSSREFS
Sequence in context: A376387 A027717 A035481 * A061214 A137024 A054264
KEYWORD
nonn
AUTHOR
Peter Luschny, Apr 01 2019
STATUS
approved