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Search: id:A125162
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| A125162 |
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a(n) = number of primes of the form k! + n. |
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+0
6
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| 1, 1, 1, 1, 3, 1, 4, 0, 1, 1, 5, 1, 3, 0, 1, 1, 6, 1, 7, 0, 1, 1, 6, 0, 1, 0, 1, 1, 6, 1, 9, 0, 0, 0, 3, 1, 11, 0, 1, 1, 9, 1, 5, 0, 1, 1, 10, 0, 2, 0, 1, 1, 9, 0, 2, 0, 1, 1, 10, 1, 9, 0, 0, 0, 3, 1, 9, 0, 1, 1, 8, 1, 9, 0, 0, 0, 5, 1, 9, 0, 1, 1, 11, 0, 1, 0, 1, 1, 8, 0, 3, 0, 0, 0, 2, 1, 10, 0, 1, 1, 10, 1
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Note the triplets of consecutive zeros in a(n) for n = {{32,33,34}, {62,63,64}, {74,75,76}, {92,93,94), {116,117,118}, {122,123,124}, {140,141,142}, {152,153,154}, {158,159,160}, {182,183,184}, {200,201,202}, {206,207,208}, {212,213,214}, {218,219,220}, {242,243,244}, {272,273,274}, {284,285,286}, ...}. The middle index of most zero triplets is a multiple of 3.
Numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1) are listed in A125164(n) = {11,21,25,31,39,41,47,51,53,61,67,69,71,73,81,91,95,99,...}. The first consecutive quintiplet of zeros has indices n = {294,295,296,297,298}, where the odd zero index n = 295 is not a multiple of 3.
Numbers n such that a(n) = 0 are listed in A125163(n) = {8, 14, 20, 24, 26, 32, 33, 34, 38, 44, 48, 50, 54, 56, 62, 63, 64, 68, 74, 75, 76, 80, 84, 86, 90, 92, 93, 94, 98, 104, 110, 114, 116, 117, 118, 120, 122, 123, 124, 128, 132, 134, 140, 141, 142, 144, 146, 152, 153, 154, 158, 159, 160, 164, 168, 170, 174, 176, 182, 183, 184, 186, 188, 194, 200, 201, 202, 204, 206, 207, 208, 212, 213, 214, 216, 218, 219, 220, 224, 230, 234, 236, 242, 243, 244, 246, 248, 252, 254, 258, 260, 264, 266, 272, 273, 274, 278, 284, 285, 286, 288, 290, 294, 295, 296, 297, 298, 300, ...} = numbers n such that no prime exists of the form k! + n.
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EXAMPLE
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a(n) is the length of n-th row in the table of numbers k such that k! + n is a prime.
The table begins: {{1}, {1}, {2}, {1}, {2, 3, 4}, {1}, {3, 4, 5, 6}, {}, {2}, {1}, {2, 3, 5, 7, 10}, {1}, {3, 4, 6}, {}, {2}, {1}, {2, 3, 4, 5, 9, 11}, {1}, {4, 5, 6, 7, 10, 11, 14}, {}, {2}, {1}, {3, 4, 6, 8, 9, 16}, {}, {3}, {}, {2}, {1}, {2, 4, 5, 11, 12, 20}, {1}, {3, 5, 6, 8, 9, 11, 17, 21, 22}, {}, {}, {}, {2, 3, 4}, {1}, {3, 4, 5, 6, 7, 8, 11, 18, 20, 21, 33}, {}, {2}, {1}, {2, 3, 6, 7, 8, 10, 14, 16, 25}, {1}, {4, 5, 12, 15, 38}, {}, {2}, {1}, {3, 4, 5, 7, 9, 10, 21, 27, 38, 44}, {}, {4, 6}, {}, ...}.
Thus a(1)-a(4) = 1, a(5) = 3, a(8) = 0, a(32)-a(34) = 0.
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MATHEMATICA
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Table[Length[Select[Range[n], PrimeQ[ #!+n]&]], {n, 1, 300}]
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CROSSREFS
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Cf. A125163 = numbers n such that no prime exists of the form k! + n. Cf. A125164 = numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1).
Sequence in context: A067009 A110790 A119719 this_sequence A123730 A143317 A130540
Adjacent sequences: A125159 A125160 A125161 this_sequence A125163 A125164 A125165
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KEYWORD
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nonn
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AUTHOR
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Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 21 2006
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