1,2

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 = Product_{q prime} (1-1/(q*(q-1)))). 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). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003

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

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

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.

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}}

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 ] & ]

Total /@ Table[PrimitiveRootList[Prime[k]], {k, 1, 45}] (* Updated for Mathematica 13 by Harlan J. Brothers, Feb 27 2023 *)

(PARI) a(n)=local(r, p, pr, j); p=prime(n); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsum(r) \\ after Franklin T. Adams-Watters's code in A060749, Michel Marcus, Mar 16 2015

Row sums of A060749, A254309.

Cf. A088145, A121380, A123475, A222009.

Sequence in context: A137095 A092097 A195295 * A100501 A291605 A142869

Adjacent sequences: A088141 A088142 A088143 * A088145 A088146 A088147

Ed Pegg Jr, Nov 03 2003

approved