cp's OEIS Frontend

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

%I A116518 #40 Mar 23 2024 20:33:24
%S A116518 1,3,15,255,65535,77805,161595,331695,575025,664335,765765,1601145,
%T A116518 2250885,2380833,2690415,3271905,3828825,4107285,5181813,5778045,
%U A116518 5871285,6007365,6613425,7448805,9258795,9787869,9935055,10503675,10554705,10724805,11060595
%N A116518 Odd numbers k such that k and phi(k) have the same number of divisors.
%C A116518 From _Farideh Firoozbakht_, Aug 28 2006: (Start)
%C A116518 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.
%C A116518 (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.
%C A116518 (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)
%C A116518 It is an open question whether this sequence contains infinitely many terms; see Bellaouar et al., 2023. - _Allen Stenger_, Feb 16 2024
%H A116518 Donovan Johnson, <a href="/A116518/b116518.txt">Table of n, a(n) for n = 1..1000</a>
%H A116518 Djamel Bellaouar, Abdelmadjid Boudaoud and Rafael Jakimczuk, <a href="https://doi.org/10.1515/ms-2023-0045">Notes on the equation d(n) = d(phi(n)) and related inequalities</a>, Math. Slovaca 73 (2023), no. 3, 613-632.
%t A116518 Select[Range[1,10510001,2],DivisorSigma[0,#]==DivisorSigma[ 0, EulerPhi[#]]&] (* _Harvey P. Dale_, Jan 30 2013 *)
%o A116518 (PARI) forstep(n=1,10^8,2,if(numdiv(n)==numdiv(eulerphi(n)),print1(n,", ")))
%Y A116518 Subsequence of A070418. Cf. A005384.
%Y A116518 Cf. A062821, A099774.
%K A116518 nonn
%O A116518 1,2
%A A116518 _Max Alekseyev_, Mar 24 2006