A241195 -id:A241195 - cp's OEIS frontend

A008330 phi(p-1), as p runs through the primes.

1, 1, 2, 2, 4, 4, 8, 6, 10, 12, 8, 12, 16, 12, 22, 24, 28, 16, 20, 24, 24, 24, 40, 40, 32, 40, 32, 52, 36, 48, 36, 48, 64, 44, 72, 40, 48, 54, 82, 84, 88, 48, 72, 64, 84, 60, 48, 72, 112, 72, 112, 96, 64, 100, 128, 130, 132, 72, 88, 96, 92, 144, 96, 120, 96, 156, 80, 96, 172, 112

Offset: 1

N. J. A. Sloane

Number of primitive roots in the field with p elements.
Kátai proves that phi(p-1)/(p-1) has a continuous distribution function. - Charles R Greathouse IV, Jul 15 2013
For odd primes p, phi(p-1)<=(p-1)/2 since p has phi(p-1) primitive roots and (p-1)/2 quadratic residues and no primitive root is a quadratic residue. - Geoffrey Critzer, Apr 18 2015

D. H. Lehmer and Emma Lehmer, "Heuristics Anyone?", in: G. Szegö et al. (eds.), Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya, Stanford University Press, 1962, pp. 202-210.

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.
S. S. Pillai, On the sum function connected with primitive roots, Proceedings of the Indian Academy of Sciences - Section A, Vol. 13. No. 6 (1941), 526-529; alternative link.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Mathematische Zeitschrift 28:1 (1928), pp. 171-199.
P. J. Stephens, An average result for Artin's conjecture, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.

Cf. A000010, A005596, A241194, A241195 (fraction phi(p-1)/(p-1)), A338364 (partial products).

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

A008330 := proc(n)
    numtheory[phi](ithprime(n)-1) ;
end proc:
seq(A008330(n),n=1..100) ;

Table[ EulerPhi[ Prime@n - 1], {n, 70}] (* Robert G. Wilson v, Dec 17 2005 *)

a(n)=eulerphi(prime(n)-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = phi(phi(prime(n))). - Robert G. Wilson v, Dec 26 2015
a(n) = phi(A006093(n)). - Michel Marcus, Dec 27 2015
Sum_{k; prime(k) <= x} a(k)/(prime(k)-1) = A * li(x) + O(x/log(x)^D), where A is Artin's constant (A005596), li(x) is the logarithmic integral, and D > 1 (Pillai, 1941; Lehmer and Lehmer 1962; Stephens, 1969). - Amiram Eldar, Jul 23 2025

A065485 Decimal expansion of Murata's constant Product_{p prime} (1 + 1/(p-1)^2).

Original entry on oeis.org

2, 8, 2, 6, 4, 1, 9, 9, 9, 7, 0, 6, 7, 5, 9, 1, 5, 7, 5, 5, 4, 6, 3, 9, 1, 7, 4, 7, 2, 3, 6, 9, 5, 3, 7, 4, 9, 0, 1, 3, 0, 4, 1, 1, 0, 5, 4, 5, 9, 2, 6, 6, 8, 7, 6, 1, 7, 9, 7, 4, 5, 8, 3, 4, 5, 3, 0, 7, 5, 7, 6, 2, 4, 4, 5, 9, 7, 6, 2, 4, 0, 5, 5, 3, 3, 4, 5, 8, 6, 6, 4, 9, 8, 8, 1, 8, 4, 4, 5

Offset: 1

N. J. A. Sloane, Nov 19 2001; edited Sep 16 2007 at the suggestion of R. J. Mathar

			2.8264199970675915755463917472369537490...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.4 and 2.7, pp. 106, 117.

Leo Murata, On the magnitude of the least prime primitive root, Journal of Number Theory, Vol. 37, No. 1 (1991), pp. 47-66.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Murata's Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A000720, A008683, A078072, A241194, A241195.

digits = 99; terms = 1000; $MaxExtraPrecision = 500; r[n_Integer] := 2 - (1-I)^(n+1) - (1+I)^(n+1); NSum[r[n-1]*PrimeZetaP[n]/n, {n, 2, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10] // Exp // RealDigits[ #, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 + 1/(p-1)^2) \\ Vaclav Kotesovec, Sep 19 2020

Equals lim_{k->oo} (1/pi(k)) * Sum_{p prime, p <= k} (p-1)/phi(p-1), where pi(k) = A000720(k) and phi(k) = A000010(k) (Murata, 1991). - Amiram Eldar, Jul 31 2020

Equals Sum_{k>=1} mu(k)^2/phi(k)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Jan 14 2022

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

Original entry on oeis.org

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

A241197 Numerator 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, 1, 1, 4, 8, 16, 288, 256, 192, 768, 384, 3456, 3072, 6912, 6144, 55296, 1658880, 221184, 110592, 3317760, 442368, 13271040

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The denominator is in A241198.

			In decimal, the minima are about 1, 0.5, 0.333333, 0.266667, 0.228571, 0.207792, 0.196856, 0.195569, 0.191808, 0.185194, 0.183469, 0.181713, 0.180525, 0.173812, 0.172676, 0.171024, 0.165507, 0.165127, 0.163588.

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}]; Numerator[Transpose[tMin][[2]]]

a(20)-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

cp's OEIS Frontend

A008330 phi(p-1), as p runs through the primes.

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A065485 Decimal expansion of Murata's constant Product_{p prime} (1 + 1/(p-1)^2).

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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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A241197 Numerator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

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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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Keywords

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Extensions