A051487 Numbers k such that phi(k) = phi(k - phi(k)).

2, 6, 12, 24, 48, 96, 150, 192, 300, 384, 600, 726, 750, 768, 1200, 1452, 1500, 1536, 2310, 2400, 2904, 3000, 3072, 3174, 3750, 4620, 4800, 5046, 5808, 5874, 6000, 6090, 6144, 6348, 6930, 7500, 7986, 9240, 9600, 10086, 10092, 10374, 11550, 11616, 11748, 12000

Offset: 1

Author

R. K. Guy

Keywords

• nonn
• nice

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.

Crossrefs

Cf. A005384, A051488.

Cf. A000010, A051953.

Programs

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

Mathematica

Select[Range[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

PARI

isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) \\ Michael B. Porter, Dec 07 2009

Extensions

More terms from James Sellers