A116518 Odd numbers k such that k and phi(k) have the same number of divisors.

1, 3, 15, 255, 65535, 77805, 161595, 331695, 575025, 664335, 765765, 1601145, 2250885, 2380833, 2690415, 3271905, 3828825, 4107285, 5181813, 5778045, 5871285, 6007365, 6613425, 7448805, 9258795, 9787869, 9935055, 10503675, 10554705, 10724805, 11060595

Offset: 1

Author

Max Alekseyev, Mar 24 2006

Keywords

• nonn

Comments

From Farideh Firoozbakht, Aug 28 2006: (Start)

For n < 6, the product of the first n Fermat primes is in the sequence because if m = 2^(2^n)-1 and n < 6 then d(m) = d(phi(m)) = 2^n.

(1). If p is a Sophie Germain prime greater than 3 then m = 69615*(2p+1) (A005385) is in the sequence because d(m) = d(phi(m)) = 96. 765765, 1601145, 3271905, 4107285, 5778045, 7448805, ... is the related subsequence.

(2). If p is a prime greater than 3 such that 4p+1 is prime then m = 700245*(4p+1) (A090866) is in the sequence because d(m) = d(phi(m)) = 160. 20307105, 37112985, 104336505, 121142385, ... is the related subsequence. (End)

It is an open question whether this sequence contains infinitely many terms; see Bellaouar et al., 2023. - Allen Stenger, Feb 16 2024

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Djamel Bellaouar, Abdelmadjid Boudaoud and Rafael Jakimczuk, Notes on the equation d(n) = d(phi(n)) and related inequalities, Math. Slovaca 73 (2023), no. 3, 613-632.

Crossrefs

Subsequence of A070418. Cf. A005384.

Cf. A062821, A099774.

Programs

Mathematica

Select[Range[1,10510001,2],DivisorSigma[0,#]==DivisorSigma[ 0, EulerPhi[#]]&] (* Harvey P. Dale, Jan 30 2013 *)

PARI

forstep(n=1,10^8,2,if(numdiv(n)==numdiv(eulerphi(n)),print1(n,", ")))