A241196 - OEIS

%I #38 Oct 13 2017 03:19:51

%S 2,3,7,31,211,2311,43891,78541,120121,870871,1381381,2282281,4084081,

%T 13123111,82192111,106696591,300690391,562582021,892371481,6915878971,

%U 71166625531,200560490131

%N Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.

%C For these p, the numerator and denominator of phi(p-1)/(p-1) are listed in A241197 and A241198. This sequence appears to be related to A073918, the smallest prime which is 1 more than a product of n distinct primes.

%C By Dirichlet's theorem on primes in arithmetic progressions, for any n there is a prime p such that p-1 is divisible by the primorial A002110(n).  Then phi(p-1)/(p-1) <= Product_{i=1..n} (1 - 1/prime(i)).  Since Sum_{i >= 1} prime(i) diverges, that goes to 0 as n -> infinity.  Thus there are primes with phi(p-1)/(p-1) arbitrarily close to 0. - _Robert Israel_, Jan 18 2016

%C 5*10^12 < a(23) <= 12234189897931. - _Giovanni Resta_, Apr 14 2016

%D R. K. Guy, Unsolved Problems in Number Theory, A2.

%H Tamiru Jarso, Tim Trudgian, <a href="https://arxiv.org/abs/1710.04320">Quadratic residues that are not primitive roots</a>, arXiv:1710.04320 [math.NT], 2017.

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

%p m:= infinity:

%p p:= 1:

%p count:= 0:

%p while count < 10 do

%p   p:= nextprime(p);

%p   r:= numtheory:-phi(p-1)/(p-1);

%p   if r < m then

%p      count:= count+1;

%p      A[count]:= p;

%p      m:= r;

%p   fi

%p od:

%p seq(A[i],i=1..count); # _Robert Israel_, Jan 18 2016

%t tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Transpose[tMin][[1]]

%Y Cf. A002110, A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

%K nonn,more

%O 1,1

%A _T. D. Noe_, Apr 17 2014

%E a(20) from _Dimitri Papadopoulos_, Jan 11 2016

%E a(21)-a(22) from _Giovanni Resta_, Apr 14 2016