A058340 Primes p such that phi(x) = p-1 has only 2 solutions, namely x = p and x = 2p.

11, 23, 29, 31, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479

Offset: 1

Author

Labos Elemer, Dec 14 2000

Keywords

• nonn

Comments

Two solutions, p and 2p, exist for all odd primes p; primes in sequence have no other solutions.

Conjecture: if q > 7 is in A005385, then q is in the sequence. - Thomas Ordowski, Jan 04 2017

Proof of conjecture: q'=(q-1)/2 is an odd prime > 3. If phi(x)=2q', which has 2-adic order 1 but is not a power of 2, there must be exactly one odd prime r dividing x. We could also have a factor of 2 (but no higher power, which would contribute more 2's to phi(x)). If x = r^e or 2r^e, then phi(x) = (r-1) r^(e-1). For this to be 2q' one possibility is r-1 = 2 and r^(e-1)=q', but then q'=r=3, ruled out by q > 7. The only other possibility is r-1=2q' and e=1, which makes r=q and x=q or 2q. - Robert Israel, Jan 04 2017

Information from Carl Pomerance: It is known that almost all primes (in the sense of relative asymptotic density) are in the sequence. - Thomas Ordowski, Jan 08 2017

Examples

			For p=2, phi(x)=1 has only two solutions, but they are 1 and 2, not 2 and 4, so 2 is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000040, A005385, A006093, A058339, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080, A138537.

Programs

Maple

filter:= n -> isprime(n) and nops(numtheory:-invphi(n-1))=2:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Aug 12 2016

Mathematica

Take[Rest@ Keys@ Select[KeySelect[KeyMap[# + 1 &, PositionIndex@ Array[EulerPhi, 10^4]], PrimeQ], Length@ # == 2 &], 54] (* Michael De Vlieger, Dec 29 2017 *)

Formula

a(n) ~ n log . - Charles R Greathouse IV, Nov 18 2022

Extensions

Edited by Ray Chandler, Jun 06 2008