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A088144
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Sum of primitive roots of n-th prime.
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| From Pieter Moree (moree(AT)mpim-bonn.mpg.de), 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.. = A005596. 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).
The number of the primitive roots is A008330(n). - R. K. Guy, Feb 25 2011
If prime(n) == 1 (mod 4), then a(n) = prime(n)*A008330(n)/2. There are also primes of the form prime(n) == 3 (mod 4) where prime(n) | a(n), namely prime(n) = 19, 127, 151, 163, 199, 251,... The list of primes in both modulo-4 classes where prime(n)|a(n) is 5, 13, 17, 19, 29, 37, 41, 53, 61,... - R. K. Guy, Feb 25 2011
a(n) = A076410(n) at n = 1, 3, 7, 55,... (for p = 2, 5, 17, 257... and perhaps only for the Fermat primes) - R. K. Guy, Feb 25 2011
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REFERENCES
| C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Leon Mirsky, The Number of Representations of an Integer as the Sum of a Prime and a k-Free Integer, Amer. Math. Monthly 56 (1949), 17-19.
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EXAMPLE
| 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
| 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
| Cf. A060749, A123475.
Sequence in context: A137095 A092097 A195295 * A100501 A142869 A086825
Adjacent sequences: A088141 A088142 A088143 * A088145 A088146 A088147
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KEYWORD
| nonn
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AUTHOR
| Ed Pegg Jr (edp(AT)wolfram.com), Nov 03 2003
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