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Search: id:A088144
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| A088144 |
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Sum of primitive roots of n-th prime. |
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+0
3
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| 1, 2, 5, 8, 23, 26, 68, 57, 139, 174, 123, 222, 328, 257, 612, 636, 886, 488, 669, 1064, 876, 1105, 1744, 1780, 1552, 2020, 1853, 2890, 1962, 2712, 2413, 3536, 4384, 3335, 5364, 3322, 3768, 4564, 7683, 7266, 8235, 4344, 8021, 6176, 8274
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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From Pieter Moree (moree(AT)science.uva.nl), Nov 03 2003: It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A denotes the Artin constant (A = prod_q (1-1/(q(q-1)), q running over all primes). Numerically A = 0.3739558136.... More precisely, Sum_{p <= x} mu(p-1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: sum_{p <= x, mu(p-1)=1} 1 = (A/2)x/log x + o(x\log x) and sum_{p <= x, mu(p-1)=-1} 1 = (A/2)x/log x + o(x/log x).
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REFERENCES
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C. F. Gauss, Disquisitiones Arithmeticae.
Leon Mirsky, Amer. Math. Monthly 56 (1949), 17-19.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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EXAMPLE
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For 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, the primitive roots are as follows: {{1}, {2}, {2, 3}, {3, 5}, {2, 6, 7, 8}, {2, 6, 7, 11}, {3, 5, 6, 7, 10, 11, 12, 14}, {2, 3, 10, 13, 14, 15}, {5, 7, 10, 11, 14, 15, 17, 19, 20, 21}, {2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27}}
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MATHEMATICA
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PrimitiveRootQ[ a_Integer, p_Integer ] := Block[ {fac, res}, fac = FactorInteger[ p - 1 ]; res = Table[ PowerMod[ a, (p - 1)/fac[ [ i, 1 ] ], p ], {i, Length[ fac ]} ]; ! MemberQ[ res, 1 ] ] PrimitiveRoots[ p_Integer ] := Select[ Range[ p - 1 ], PrimitiveRootQ[ #, p ] & ]
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CROSSREFS
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Sequence in context: A009735 A137095 A092097 this_sequence A100501 A142869 A086825
Adjacent sequences: A088141 A088142 A088143 this_sequence A088145 A088146 A088147
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KEYWORD
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nonn
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AUTHOR
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Ed Pegg Jr (edp(AT)wolfram.com), Nov 03 2003
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