A060473 -id:A060473 - cp's OEIS frontend

A203966 Numbers n such that phi(n) divides n+1, where phi is Euler's totient function (A000010).

1, 2, 3, 15, 255, 65535, 83623935, 4294967295, 6992962672132095

Offset: 1

José María Grau Ribas, Jan 08 2012

Numbers k such that A060473(k) = 1. - Robert G. Wilson v, Jul 06 2014
Except for a(2), all terms are odd. - Chai Wah Wu, Jun 06 2017
Since gcd(phi(n),n) = 1, all terms are squarefree. Then, for n = p1 * ... * pk with primes p1 < ... < pk, (n+1)/phi(n) is very close to p1/(p1-1)*...*pk/(p1-1). Since p/(p-1) is decreasing as p grows, having (n+1)/phi(n) = 3 implies that p1 >= 5 and further that n >= 2.4*10^56 is a product of at least 33 primes. Similarly, having (n+1)/phi(n) >= 4 implies that n >= 1.6*10^30 is a product of at least 21 primes. Hence, the terms of this sequence below 1.6*10^30 have (n+1)/phi(n) = 2 and thus coincide with A050474. - Max Alekseyev, Jan 30 2022

			15 is in the sequence because phi(15) = 8, and 8 divides 16 = 15 + 1 evenly.

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.

Cf. A060473, A000010, A202855.

Superset of A050474.

Select[Range[100000], Divisible[#+1, EulerPhi[#]]&]

a(8) from Donovan Johnson, Jan 13 2012

a(9) confirmed by Max Alekseyev, Jan 30 2022

A060474 a(n) = denominator of phi(n)/(n+1), where phi(n) is Euler's phi, A000010.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 5, 2, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 9, 14, 29, 15, 31, 16, 33, 17, 35, 3, 37, 19, 13, 5, 41, 21, 43, 22, 9, 23, 47, 24, 49, 25, 51, 13, 53, 27, 55, 7, 19, 29, 59, 30, 61, 31, 21, 16, 65, 11, 67, 34, 69, 35, 71, 36, 73, 37, 25, 19, 77, 13, 79, 40

Offset: 1

Fabian Rothelius, Mar 16 2001

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000010, A060473.

with(numtheory,phi): seq(denom(phi(n)/(n+1)), n=1..50);

Denominator[Table[EulerPhi[n]/(n+1),{n,80}]] (* Harvey P. Dale, Apr 13 2012 *)

{ for (n=1, 1000, write("b060474.txt", n, " ", denominator(eulerphi(n)/(n + 1))); ) } \\ Harry J. Smith, Jul 13 2009

from sympy import totient, gcd
def A060474(n): return (n+1)//gcd(n+1,totient(n)) # Chai Wah Wu, Apr 02 2021

More terms from Asher Auel, Mar 16 2001

A Maple program that should have been a PARI program removed by Harry J. Smith, Jul 13 2009

cp's OEIS Frontend

A203966 Numbers n such that phi(n) divides n+1, where phi is Euler's totient function (A000010).

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