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A300629 a(1) = 561; a(n+1) = smallest Fermat pseudoprime to all natural bases up to lpf(a(n)). 4
561, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, 413138881, 2301745249, 2438403661, 5255104513, 5781222721, 8251854001, 12173703001, 13946829751, 15906120889, 23224518901, 31876135201, 51436355851, 57274147841, 58094662081 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It is sufficient to consider only prime bases: a(n+1) is the least composite number k such that p^(k-1) == 1 (mod k) for every prime p <= lpf(a(n)), with a(1) = 561.

Conjecture: a(n+1) is the smallest Carmichael number k such that lpf(k) > lpf(a(n)), with a(1) = 561. It seems that such Carmichael numbers have exactly three prime factors.

The above conjecture is true if A083876(n) < A285549(n) for all n > 1, but has not been proven; there is no a counterexample up to a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018

Carl Pomerance (in a letter to the author) wrote, Mar 13 2018: (Start)

  Assuming a strong form of the prime k-tuples conjecture, if there are no small counterexamples, there are likely to be none.

  Here's why.

  Assuming prime k-tuples, there are infinitely many Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where each factor is prime. And from Bateman--Horn, these are fairly thickly distributed. There are other even better triples such as (60k+41)(90k+61)(150k+101), where "better" means the least prime factor is not so far below the cube root.

  So, to get into the sequence, a number needs to be a Fermat pseudoprime for every base up to nearly the cube root.

  However, it's a theorem that a sufficiently large number with this property must be a Carmichael number. (end)

Theorem: if lpf(a(n)) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= lpf(a(n)). - Thomas Ordowski, Mar 13 2018

lpf(a(n)) are listed in A300748. - Max Alekseyev, Mar 13 2018

For m > 1, A135720(m) >= A083876(m-1), with equality iff lpf(a(n)) = prime(m); by this conjecture in the second comment. - Thomas Ordowski, Mar 13 2018

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..138

CROSSREFS

Cf. A002997, A020639, A083876, A271221, A285549, A300748.

Subsequence of A087788 and of A135720.

Sequence in context: A087788 A173703 A306338 * A135720 A263403 A083733

Adjacent sequences:  A300626 A300627 A300628 * A300630 A300631 A300632

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Mar 10 2018

STATUS

approved

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Last modified January 28 06:29 EST 2020. Contains 331317 sequences. (Running on oeis4.)