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A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1). 1
1, 1, 5, 2, 19, 5, 41, 4, 15, 19, 109, 10, 155, 41, 95, 8, 271, 15, 341, 38, 205, 109, 505, 20, 95, 155, 45, 82, 811, 95, 929, 16, 545, 271, 779, 30, 1331, 341, 775, 76, 1639, 205, 1805, 218, 285, 505, 2161, 40, 287, 95, 1355, 310, 2755, 45, 2071, 164, 1705, 811 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For any positive integer n and any m coprime to n, define R(n,m) = Product_{primes p divides n} (p - [m == 1 (mod p)]), where [] is an Iverson branket. Then we have the following conjecture: (Start)

Let k == 2, 3 (mod 4) be a squarefree number, b be any positive integer such that k*b^2 is not a perfect power and not equal to -1, n be either coprime to or divisible by 4*k. Define Q(N,k*b^2,n,m) = # {primes p <= N : p == m (mod n), k*b^2 is a primitive modulo p}, then:

(a) if gcd(n, 4*k) = 1, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ R(n,m)/a(n);

(b) if 4*k divides n, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ 2*R(n,m)/a(n) if Jacobi(k/m) = -1 and 0 if Jacobi(k/m) = +1,

Where C is the Artin's constant = A005596, PrimePi = A000720. (End)

(Note that Sum_{m=1..n, gcd(m,n)=1} R(n,m) = a(n).)

For example, let N = 10^6:

k*b^2 |  n |  m | Q(N,k*b^2,n,m) | Q(N,k*b^2,n,m)/(C*PrimePi(N))

   2  |  8 |  3 |      14642     |   0.498794... approx = 2/4

   3  |  5 |  1 |       6192     |   0.210936... approx = 4/19

  -2  | 48 | 13 |       2933     |   0.099915... approx = 4/40

  -5  |  9 |  5 |       5933     |   0.202113... approx = 3/15

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Artin's constant

Wikipedia, Artin's conjecture on primitive roots

MAPLE

P := (p, e) -> p^(e-1)*(p^2 - p - 1):

a := n -> mul(P(f[1], f[2]), f in ifactors(n)[2]):

seq(a(n), n=1..58); # Peter Luschny, Feb 13 2019

MATHEMATICA

a[n_] := Product[{p, e} = pe; p^(e-1) (p^2-p-1), {pe, FactorInteger[n]}]; a[1] = 1; Array[a, 58] (* Jean-Fran├žois Alcover, Jul 22 2019 *)

PROG

(PARI) a(n) = my(f=factor(n)); prod(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); (p^2 - p - 1)*p^(e-1))

CROSSREFS

Cf. A000720 (PrimePi), A005596 (Artin's constant).

Sequence in context: A286252 A286154 A304635 * A276533 A303685 A189746

Adjacent sequences:  A306195 A306196 A306197 * A306199 A306200 A306201

KEYWORD

easy,nonn,mult

AUTHOR

Jianing Song, Jan 28 2019

STATUS

approved

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Last modified September 15 04:09 EDT 2019. Contains 327062 sequences. (Running on oeis4.)