cp's OEIS Frontend

A241196 Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.

%I A241196 #39 Feb 16 2025 08:33:21
%S A241196 2,3,7,31,211,2311,43891,78541,120121,870871,1381381,2282281,4084081,
%T A241196 13123111,82192111,106696591,300690391,562582021,892371481,6915878971,
%U A241196 71166625531,200560490131
%N A241196 Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.
%C A241196 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 A241196 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 A241196 5*10^12 < a(23) <= 12234189897931. - _Giovanni Resta_, Apr 14 2016
%D A241196 R. K. Guy, Unsolved Problems in Number Theory, A2.
%H A241196 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 A241196 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EuclidNumber.html">Euclid Number</a>
%p A241196 m:= infinity:
%p A241196 p:= 1:
%p A241196 count:= 0:
%p A241196 while count < 10 do
%p A241196   p:= nextprime(p);
%p A241196   r:= numtheory:-phi(p-1)/(p-1);
%p A241196   if r < m then
%p A241196      count:= count+1;
%p A241196      A[count]:= p;
%p A241196      m:= r;
%p A241196   fi
%p A241196 od:
%p A241196 seq(A[i],i=1..count); # _Robert Israel_, Jan 18 2016
%t A241196 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 A241196 Cf. A002110, A008330 (phi(prime(n)-1)), A073918, A241194, A241195.
%K A241196 nonn,more
%O A241196 1,1
%A A241196 _T. D. Noe_, Apr 17 2014
%E A241196 a(20) from _Dimitri Papadopoulos_, Jan 11 2016
%E A241196 a(21)-a(22) from _Giovanni Resta_, Apr 14 2016