login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).
1

%I #33 Dec 01 2022 10:53:43

%S 1,1,5,2,19,5,41,4,15,19,109,10,155,41,95,8,271,15,341,38,205,109,505,

%T 20,95,155,45,82,811,95,929,16,545,271,779,30,1331,341,775,76,1639,

%U 205,1805,218,285,505,2161,40,287,95,1355,310,2755,45,2071,164,1705,811

%N Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

%C 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)

%C 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:

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

%C (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,

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

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

%C For example, let N = 10^6:

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

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

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

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

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

%H Jianing Song, <a href="/A306198/b306198.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/18) * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 1/p^4) = 0.1314639252... . - _Amiram Eldar_, Dec 01 2022

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

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

%p seq(a(n), n=1..58); # _Peter Luschny_, Feb 13 2019

%t a[n_] := Product[{p, e} = pe; p^(e-1) (p^2-p-1), {pe, FactorInteger[n]}]; a[1] = 1; Array[a, 58] (* _Jean-François Alcover_, Jul 22 2019 *)

%o (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))

%Y Cf. A000720 (PrimePi), A005596 (Artin's constant), A086463.

%K easy,nonn,mult

%O 1,3

%A _Jianing Song_, Jan 28 2019