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A300629
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a(1) = 561; a(n+1) = smallest Fermat pseudoprime to all natural bases up to lpf(a(n)).
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5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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
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
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LINKS
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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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