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A328935 Imprimitive Carmichael numbers: Carmichael numbers m such that if m = p_1 * p_2 * ... *p_k is the prime factorization of m then g(m) = gcd(p_1 - 1, ..., p_k - 1) > sqrt(lambda(m)), where lambda is the Carmichael lambda function (A002322). 4
294409, 399001, 488881, 512461, 1152271, 1461241, 3057601, 3828001, 4335241, 6189121, 6733693, 10267951, 14676481, 17098369, 19384289, 23382529, 50201089, 53711113, 56052361, 64377991, 68154001, 79624621, 82929001, 84350561, 96895441, 115039081, 118901521, 133800661 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Granville and Pomerance separated the Carmichael numbers into two classes, primitive and imprimitive, according to whether g(m) <= sqrt(lambda(n)) or not.

They conjectured that most Carmichael numbers are primitive and most 3-Carmichael numbers (A087788) are imprimitive.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

FORMULA

Terms m of A002997 such that A258409(m) > sqrt(A002322(m)).

MATHEMATICA

aQ[n_] := Length[(f = FactorInteger[n])] > 2 && Max[f[[;; , 2]]] == 1 && Divisible[n-1, (lambda = LCM @@ (f[[;; , 1]] - 1))] && GCD @@ (f[[;; , 1]] - 1) > Sqrt[lambda]; Select[Range[4*10^6], aQ]

CROSSREFS

Cf. A002322, A002997, A087788, A258409.

Sequence in context: A050249 A224973 A328664 * A182206 A178997 A328938

Adjacent sequences:  A328932 A328933 A328934 * A328936 A328937 A328938

KEYWORD

nonn

AUTHOR

Amiram Eldar, Oct 31 2019

STATUS

approved

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Last modified January 24 01:05 EST 2020. Contains 331178 sequences. (Running on oeis4.)