OFFSET
1,1
COMMENTS
This sequence is infinite, in fact 3*2^n is a subsequence because if m = 3*2^n then phi(m-phi(m)) = phi(3*2^n-2^n) = 2^n = phi(m). Also, if p is a Sophie Germain prime greater than 3 then for each natural number n, 2^n*3*p^2 is in the sequence. Note that there exist terms of this sequence like 750 or 2310 that they aren't of either of these forms. - Farideh Firoozbakht, Jun 19 2005
If n is an even term greater than 2 in this sequence then 2n is also in the sequence. Because for even numbers m we have phi(2m) = 2*phi(m) so phi(2n) = 2*phi(n) = 2*phi(n-phi(n)) and since n is an even number greater than 2, n-phi(n) is even so 2*phi(n-phi(n)) = phi(2n-2*phi(n)) = phi(2n-phi(2n)) hence phi(2n) = phi(2n-phi(2n)) and 2n is in the sequence. Conjecture: All terms of this sequence are even. - Farideh Firoozbakht, Jul 04 2005
If n is in the sequence and the natural number m divides gcd(n,phi(n)) then m*n is in the sequence. The facts that I have found about this sequence earlier (Jun 19 2005 and Jul 04 2005) are consequences of this. If p is a Sophie Germain prime greater than 3, k>1 and k & n are natural numbers then 2^n*3*p^k are in the sequence. - Farideh Firoozbakht, Dec 10 2005
Numbers n such that phi(n) = phi(n + phi(n)) includes all n = 2^k. - Jonathan Vos Post, Oct 25 2007
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
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[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)
PROG
(PARI) isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) \\ Michael B. Porter, Dec 07 2009
(Haskell)
a051487 n = a051487_list !! (n-1)
a051487_list = [x | x <- [2..], let t = a000010 x, t == a000010 (x - t)]
-- Reinhard Zumkeller, Jun 03 2013
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved