A241197 -id:A241197 - cp's OEIS frontend

A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 4, 1, 2, 2, 11, 6, 14, 4, 10, 12, 1, 4, 20, 5, 1, 2, 16, 26, 1, 3, 2, 24, 8, 22, 18, 4, 4, 1, 41, 21, 44, 4, 36, 1, 3, 10, 8, 12, 56, 6, 14, 48, 4, 2, 1, 65, 33, 4, 22, 12, 46, 36, 16, 12, 4, 39, 8, 2, 86, 28, 5, 89, 20, 10, 2, 95

Offset: 1

T. D. Noe, Apr 17 2014

The denominators are in A241195. The new minima of phi(p-1)/(p-1) occur at primes listed in A241196. The numerator and denominator of those terms are in A241197 and A241198.

For primes p>2, the fraction phi(p - 1)/(p - 1) has the maximum value = 1/2 if and only if p is in A019434. - Geoffrey Critzer, Dec 30 2014

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 117.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of some sequences of numbers, III., J. London Math. Soc. 13 (1938), pp. 119-127.
Imre Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), pp. 278-289.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.

Cf. A000010, A005596, A008330 (phi(prime(n)-1)), A065485, A241195, A241196, A241197, A241198.

Cf. A076512, A006093.

[Numerator(EulerPhi(NthPrime(n)-1)/(NthPrime(n)-1)): n in [1..80]]; // Vincenzo Librandi, Apr 06 2015

seq(numer(numtheory:-phi(ithprime(i)-1)/(ithprime(i)-1)), i=1..100); # Robert Israel, Jan 11 2015

Numerator[Table[EulerPhi[p - 1]/(p - 1), {p, Prime[Range[100]]}]]

lista(nn) = forprime(p=2, nn, print1(numerator(eulerphi(p-1)/(p-1)), ", ")); \\ Michel Marcus, Jan 03 2015

From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A241195(n) = 0.373955... (Artin's constant, A005596).
Asymptotic mean of inverse ratio: lim_{m->oo} (1/m) * Sum_{n=1..m} A241195(n)/a(n) = 2.826419... (Murata's constant, A065485). (End)
a(n) = A076512(A006093(n)). - Ridouane Oudra, Mar 24 2025

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

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Apr 17 2014

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

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

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

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

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

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]]

a(20) from Dimitri Papadopoulos, Jan 11 2016

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

A241198 Denominator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 2, 3, 15, 35, 77, 1463, 1309, 1001, 4147, 2093, 19019, 17017, 39767, 35581, 323323, 10023013, 1339481, 676039, 20957209, 2800733, 86822723

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The numerator is in A241197.

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

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}]; Denominator[Transpose[tMin][[2]]]

a(20)-a(22) from Giovanni Resta, Apr 14 2016

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A241194 Numerator of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

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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.

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A241198 Denominator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

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