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 A305237 Numbers n such that n, n+1 and n+2 all have primitive roots. 1
 1, 2, 3, 4, 5, 9, 17, 25, 81, 241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Start of run of 3 consecutive numbers in A033948. The next term is 3^541 - 2, which is too large to be included here. No more terms below 3^100000, or approximately 1.33*10^47712. There is a multiple of 4 in every four consecutive positive integers and it clearly has no primitive roots if it is larger than 4. Again, there is a multiple of 3 in every three consecutive positive integers, so it must be a power of 3 or two times a power of 3, and the other two numbers must be odd prime powers or two times odd prime powers. According to Pillai's conjecture, there're only finitely many solutions to |3^a - p^b| = 2, |3^a - 2*p^b| = 1, |p^a - 2*3^b| = 1 with a,b >= 2, p odd primes (no solution other than 3^3 - 5^2 = 2, 3^5 - 2*11^2 = 1 below 3^100000). So beyond (25, 26, 27) and (241, 242, 243), it's very likely that all three consecutive numbers with primitive roots are of the form (3^i, 3^i + 1, 3^i + 2), (3^j - 2, 3^j - 1, 3^j), (2*3^k - 1, 2*3^k, 2*3^k + 1) such that (3^i + 1)/2, 3^i + 2, 3^j - 2, (3^j - 1)/2, 2*3^k - 1, 2*3^k + 1 are primes, which only produces one more solution (3^541 - 2, 3^541 - 1, 3^541) below 3^1000000. LINKS EXAMPLE 81, 82, 83 all have primitive roots (in fact, their least common primitive root is 47), so 81 is a term. Note that A014224 and A028491 have a term 541 in common, so 3^541 - 2, 3^541 - 1 and 3^541 all have primitive roots, so 3^541 - 2 is a term. CROSSREFS Cf. A003306, A003307, A014224, A028491, A051783, A171381. Sequence in context: A003271 A049796 A106165 * A088817 A018896 A162374 Adjacent sequences:  A305234 A305235 A305236 * A305238 A305239 A305240 KEYWORD nonn,hard,more AUTHOR Jianing Song, Jun 04 2018 STATUS approved

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Last modified December 1 02:46 EST 2021. Contains 349426 sequences. (Running on oeis4.)