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

2, 3, 7, 31, 211, 2311, 43891, 78541, 120121, 870871, 1381381, 2282281, 4084081, 13123111, 82192111, 106696591, 300690391, 562582021, 892371481, 6915878971, 71166625531, 200560490131

Offset: 1

Author

T. D. Noe, Apr 17 2014

Keywords

• nonn
• more

Comments

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.

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

5*10^12 < a(23) <= 12234189897931. - Giovanni Resta, Apr 14 2016

References

• R. K. Guy, Unsolved Problems in Number Theory, A2.

Links

• Tamiru Jarso, Tim Trudgian, Quadratic residues that are not primitive roots, arXiv:1710.04320 [math.NT], 2017.
• Eric Weisstein's World of Mathematics, Euclid Number

Crossrefs

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

Programs

Maple

m:= infinity:
p:= 1:
count:= 0:
while count < 10 do
  p:= nextprime(p);
  r:= numtheory:-phi(p-1)/(p-1);
  if r < m then
     count:= count+1;
     A[count]:= p;
     m:= r;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Jan 18 2016

Mathematica

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]]

Extensions

a(20) from Dimitri Papadopoulos, Jan 11 2016

a(21)-a(22) from Giovanni Resta, Apr 14 2016