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 A289237 Find the first (maximal) string, of length exactly n, of consecutive primes that alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1. The first element is a(n). 2
 53, 29, 67, 37, 449, 179, 5, 389, 89, 2213, 11149, 10369, 6761, 113341, 80447, 151909, 43777, 2964553, 1457333, 175573, 809, 3954889, 121930481, 96050953, 15186319, 296080717, 98380549, 77011289, 2720227693, 5696814287, 1572386903, 4136299357, 288413159 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS By the first Formula, a(21) = 809 since A289119(21) = 809 < A289119(22). REFERENCES R. K. Guy, Unsolved Problems in Number Theory, A4. LINKS Giovanni Resta, Table of n, a(n) for n = 1..43 Jens Kruse Andersen, Consecutive Congruent Primes FORMULA a(n) = A289119(n) if and only if n > 1 and A289119(n) < A289119(n+1). EXAMPLE {Prime[k], Mod[Prime[k], 6]} = {2, 2}, {3, 3}, {5, 5}, {7, 1}, {11, 5}, {13, 1}, {17, 5}, {19, 1}, {23, 5}, {29, 5}, {31, 1}, {37, 1}, {41, 5}, {43, 1}, {47, 5}, {53, 5}, {59, 5}, {61, 1}, {67,  1}, {71, 5}, {73, 1}, {79, 1}, . ., so a(n) = 53, 29, 67, 37 for n = 1, 2, 3, 4 and a(7) = 5. MATHEMATICA i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 6] != 0 || Mod[ Prime[i + 1] - Prime[i], 6] != 0, i++]; T = {Prime[i]}; Do[j = 3; While[ ! (Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] != 0 && (Mod[ Prime[j] - Prime[j - 1], 6] == 0 || j == 3) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 6] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 16}]; T CROSSREFS Cf. A247384, A289118, A289119. Sequence in context: A048602 A048606 A033373 * A319904 A171132 A259091 Adjacent sequences:  A289234 A289235 A289236 * A289238 A289239 A289240 KEYWORD nonn AUTHOR Jonathan Sondow, Jun 28 2017 EXTENSIONS a(19)-a(33) from Giovanni Resta, Jun 29 2017 STATUS approved

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Last modified December 9 03:23 EST 2021. Contains 349625 sequences. (Running on oeis4.)