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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..37.

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

Cf. A002997, A322937.

Sequence in context: A056831 A027717 A035481 * A061214 A137024 A054264

Adjacent sequences:  A323211 A323212 A323213 * A323215 A323216 A323217

KEYWORD

nonn

AUTHOR

Peter Luschny, Apr 01 2019

STATUS

approved

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Last modified January 16 18:17 EST 2022. Contains 350376 sequences. (Running on oeis4.)