A241196 -id:A241196 - 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

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

A293605 Primes p such that phi(p-1) < (p-1)/4.

Original entry on oeis.org

211, 331, 421, 631, 661, 991, 1051, 1171, 1321, 1471, 1951, 2311, 2341, 2521, 2731, 2971, 3121, 3301, 3361, 3511, 3571, 3631, 4201, 4621, 4831, 4951, 5281, 5851, 5881, 6007, 6091, 6271, 6301, 7351, 7411, 7561, 7591, 8191, 8581, 8779, 8821, 8971, 9241, 9283, 9661, 9871, 9901

Offset: 1

Michel Marcus, Oct 13 2017

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tamiru Jarso and Tim Trudgian, Quadratic non-residues that are not primitive roots, Mathematics of Computation, Vol. 88, No. 317 (2019), pp. 1251-1260; arXiv preprint, arXiv:1710.04320 [math.NT], 2017.

Cf. A000010, A008330, A241196.

Select[Prime[Range[1500]],EulerPhi[#-1]<(#-1)/4&] (* Harvey P. Dale, Oct 27 2018 *)

lista(nn) = forprime(p=2, nn, if (eulerphi(p-1) < (p-1)/4, print1(p, ", ")));

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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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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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Comments

Crossrefs

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Mathematica

Extensions

A293605 Primes p such that phi(p-1) < (p-1)/4.

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Author

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Crossrefs

Programs

Mathematica

PARI