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A116518
Odd numbers k such that k and phi(k) have the same number of divisors.
3
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
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.
Sequence in context: A257018 A173146 A139289 * A247174 A277626 A050474
KEYWORD
nonn
AUTHOR
Max Alekseyev, Mar 24 2006
STATUS
approved