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A323214
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Composite numbers k such that p^(k-1) == 1 (mod k) for every prime p strongly prime to k.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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EXAMPLE
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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).
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PROG
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(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
return [n for n in (1..len) if is_strongCarmichael(n)]
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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