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Numbers k such that phi(k) = phi(k - phi(k)).
7

%I #40 Feb 19 2024 01:58:22

%S 2,6,12,24,48,96,150,192,300,384,600,726,750,768,1200,1452,1500,1536,

%T 2310,2400,2904,3000,3072,3174,3750,4620,4800,5046,5808,5874,6000,

%U 6090,6144,6348,6930,7500,7986,9240,9600,10086,10092,10374,11550,11616,11748,12000

%N Numbers k such that phi(k) = phi(k - phi(k)).

%C 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

%C 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

%C 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

%C Numbers n such that phi(n) = phi(n + phi(n)) includes all n = 2^k. - _Jonathan Vos Post_, Oct 25 2007

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

%H Amiram Eldar, <a href="/A051487/b051487.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, <a href="https://www.researchgate.net/profile/Marek-Wojtowicz-2/publication/266273753_A_conjecture_of_Erdos_concerning_inequalities_for_the_Euler_totient_function">A conjecture of Erdős concerning inequalities for the Euler totient function</a>, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.

%t Select[Range[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (* _Farideh Firoozbakht_, Jun 19 2005 *)

%o (PARI) isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) \\ _Michael B. Porter_, Dec 07 2009

%o (Haskell)

%o a051487 n = a051487_list !! (n-1)

%o a051487_list = [x | x <- [2..], let t = a000010 x, t == a000010 (x - t)]

%o -- _Reinhard Zumkeller_, Jun 03 2013

%Y Cf. A005384, A051488.

%Y Cf. A000010, A051953.

%K nonn,nice

%O 1,1

%A _R. K. Guy_

%E More terms from _James A. Sellers_