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A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).

%I A306198 #34 Feb 16 2025 08:33:55
%S A306198 1,1,5,2,19,5,41,4,15,19,109,10,155,41,95,8,271,15,341,38,205,109,505,
%T A306198 20,95,155,45,82,811,95,929,16,545,271,779,30,1331,341,775,76,1639,
%U A306198 205,1805,218,285,505,2161,40,287,95,1355,310,2755,45,2071,164,1705,811
%N A306198 Multiplicative with a(p^e) = p^(e-1)*(p^2 - p - 1).
%C A306198 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 A306198 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 A306198 (a) if gcd(n, 4*k) = 1, then Q(N,k*b^2,n,m)/(C*PrimePi(N)) ~ R(n,m)/a(n);
%C A306198 (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 A306198 Where C is the Artin's constant = A005596, PrimePi = A000720. (End)
%C A306198 (Note that Sum_{m=1..n, gcd(m,n)=1} R(n,m) = a(n).)
%C A306198 For example, let N = 10^6:
%C A306198 k*b^2 |  n |  m | Q(N,k*b^2,n,m) | Q(N,k*b^2,n,m)/(C*PrimePi(N))
%C A306198    2  |  8 |  3 |      14642     |   0.498794... approx = 2/4
%C A306198    3  |  5 |  1 |       6192     |   0.210936... approx = 4/19
%C A306198   -2  | 48 | 13 |       2933     |   0.099915... approx = 4/40
%C A306198   -5  |  9 |  5 |       5933     |   0.202113... approx = 3/15
%H A306198 Jianing Song, <a href="/A306198/b306198.txt">Table of n, a(n) for n = 1..10000</a>
%H A306198 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArtinsConstant.html">Artin's constant</a>.
%H A306198 Wikipedia, <a href="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots">Artin's conjecture on primitive roots</a>.
%F A306198 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 A306198 P := (p, e) -> p^(e-1)*(p^2 - p - 1):
%p A306198 a := n -> mul(P(f[1], f[2]), f in ifactors(n)[2]):
%p A306198 seq(a(n), n=1..58); # _Peter Luschny_, Feb 13 2019
%t A306198 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 A306198 (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 A306198 Cf. A000720 (PrimePi), A005596 (Artin's constant), A086463.
%K A306198 easy,nonn,mult
%O A306198 1,3
%A A306198 _Jianing Song_, Jan 28 2019