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A287862 Numbers n with a record size of the largest Carmichael number that can be generated from them using Erdős's method. 1
36, 60, 108, 112, 120, 180, 216, 360, 540, 840, 1200, 1620, 2016, 2160, 2520, 3360, 3780, 4800, 5040, 6480, 7560, 8400, 10080, 12600, 15120, 25200 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Erdős showed in 1956 how to construct Carmichael numbers from a given number n (typically with many divisors). Given a number n, let P be the set of primes p such that (p-1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == 1 (mod n) then c is a Carmichael number.
The corresponding largest Carmichael numbers are 63973, 172081, 188461, 278545, 852841, 31146661, 509033161, 416937760921, ...
LINKS
Paul Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), pp. 201-206.
Andrew Granville, Primality testing and Carmichael numbers, Notices of the American Mathematical Society, Vol. 39 No. 6 (1992), pp. 696-700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.
EXAMPLE
The set of primes for n = 36 is P={5, 7, 13, 19, 37}. Two subsets, {7, 13, 19} and {7, 13, 19, 37} have c == 1 (mod n): c = 7*13*19 = 1729 and c = 7*13*19*37 = 63973. 36 is the first number that generates Carmichael numbers thus a(1)=36.
MATHEMATICA
a = {}; cmax = 0; Do[p = Select[Divisors[n] + 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c = Max[c, c1]], {j, 1, Length[ps]}];
If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a
CROSSREFS
Sequence in context: A188633 A328961 A335295 * A066505 A039419 A043242
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Sep 01 2017
STATUS
approved

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Last modified June 18 17:48 EDT 2024. Contains 373483 sequences. (Running on oeis4.)