OFFSET
1,1
COMMENTS
If p is a Sophie Germain prime greater than 3 and n is a natural number then 2^n*3*p is in the sequence. That is because if m = 2^n*3*p then phi(m) = 2^n*(p-1) and phi(m - phi(m)) = phi(2^n*3*p - 2^n*(p-1)) = phi(2^n*(2p+1)) = 2^n*p so phi(m) < phi(m-phi(m)) and m is in the sequence. - Farideh Firoozbakht, Jun 19 2005
Erdős (1980) proposed the problem to prove that this sequence is infinite and has an asymptotic density 0. Grytczuk et al. (2001) proved that this sequence is infinite with an upper asymptotic density < 0.45637. - Amiram Eldar, May 22 2021
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, A conjecture of Erdős concerning inequalities for the Euler totient function, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.
MATHEMATICA
Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)
PROG
(Haskell)
a051488 n = a051488_list !! (n-1)
a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)]
-- Reinhard Zumkeller, Apr 12 2014
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved