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A241196 Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function. 4
2, 3, 7, 31, 211, 2311, 43891, 78541, 120121, 870871, 1381381, 2282281, 4084081, 13123111, 82192111, 106696591, 300690391, 562582021, 892371481, 6915878971, 71166625531, 200560490131 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..22.

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

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]]

CROSSREFS

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

Sequence in context: A046972 A006862 A038710 * A073918 A096350 A018239

Adjacent sequences:  A241193 A241194 A241195 * A241197 A241198 A241199

KEYWORD

nonn,more

AUTHOR

T. D. Noe, Apr 17 2014

EXTENSIONS

a(20) from Dimitri Papadopoulos, Jan 11 2016

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

STATUS

approved

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Last modified June 13 11:32 EDT 2021. Contains 344990 sequences. (Running on oeis4.)