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

1,2

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.

Mathematica

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

Prog

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

Crossrefs

Subsequence of A070418. Cf. A005384.

Cf. A062821, A099774.

Sequence in context: A173146 A139289 A250405 * A247174 A277626 A050474

Adjacent sequences: A116515 A116516 A116517 * A116519 A116520 A116521

Keyword

nonn

Author

Max Alekseyev, Mar 24 2006

Status

approved