

A050474


Solutions to 2*phi(x) = x+1.


8




OFFSET

1,2


COMMENTS

If n is in the sequence and n+2 is prime then m=n*(n+2) is in the sequence because 2*phi(m) = 2*phi(n*(n+2)) = 2*phi(n)*(n+1) = (n+1)^2 = m+1. We can obtain the terms 3, 15, 255, 65535 & 4294967295 from 1 (the first term) in this way. Also since 83623935 is a term and 83623935+2 is prime 83623935*(83623935+2)=6992962672132095 is in the sequence. So 1 and 83623935 are the only known independent terms and next term of this sequence if it exists is the third such term.  Farideh Firoozbakht, May 01 2007
The next term, if it exists, has at least 7 distinct prime factors (see Beiler, p. 92).  Jud McCranie, Dec 13 2012
From Chris Boyd, Mar 22 2015: (Start)
Solutions to k*phi(x) = x + 1, including a(1)  a(8), were published in 1932 by D. H. Lehmer. In the paper's summing up, "3*5*353*929" (= 4919055) was printed in error; it should have read "3*5*17*353*929" (= 83623935), i.e., a(6). This error has been propagated in several subsequent texts, including Wong's thesis.
Lehmer identified solutions where x has fewer than 7 distinct prime factors. Wong showed that no additional solutions exist unless x has at least 8 distinct prime factors. It appears not to be excluded by either author that an unidentified solution < a(8) with 8 or more distinct prime factors may exist. (End)


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, page 92.


LINKS

Table of n, a(n) for n=1..8.
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745751.
E. Wong, Computations on normal families of primes, Simon Fraser University, 1997, MSc thesis.


FORMULA

A number n is in the sequence iff phi(n^2)=1+2+3+...+n because n is in the sequence <=> 2*phi(n)=n+1 <=> n*phi(n)=n*(n+1)/2 <=> phi(n^2)=1+2+3++...+n. For n=1,2,...,5, a(n)=2^2^(n1)1.  Farideh Firoozbakht, Jan 26 2006


EXAMPLE

2*phi(15) = 2*8 = 15 + 1, so 15 is a member of the sequence.


MAPLE

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if 2*phi(n)=(n+1) then print(n); fi;
od; end: P(10^20); # Paolo P. Lava, May 29 2014


MATHEMATICA

Select[Range[700000], (# + 1)== 2 EulerPhi[#] &] (* Vincenzo Librandi, Mar 22 2015 *)


PROG

(PARI) is_A050474(n)=if(2*eulerphi(n)==n+1, 1, 0) \\ Chris Boyd, Mar 22 2015
(MAGMA) [n: n in [1..2*10^6]  2*EulerPhi(n) eq (n+1)]; // Vincenzo Librandi, Mar 22 2015


CROSSREFS

Cf. A000010, A129613, A129614, A129615, A202855, A203966.
Sequence in context: A116518 A247174 A277626 * A051179 A122591 A326263
Adjacent sequences: A050471 A050472 A050473 * A050475 A050476 A050477


KEYWORD

hard,nonn


AUTHOR

Jud McCranie, Dec 24 1999


STATUS

approved



