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Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.
4

%I #16 Nov 15 2024 07:00:09

%S 1,2,8,10,20,22,28,30,32,44,46,48,52,54,56,58,66,70,72,78,82,92,96,

%T 102,104,106,110,116,120,126,130,132,136,138,140,148,150,156,164,166,

%U 172,178,190,196,198,204,210,212,216,220,222,226,228,238,240,250,260,262

%N Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.

%C All terms except 1 are even. - _Robert Israel_, Mar 29 2020

%H Robert Israel, <a href="/A071388/b071388.txt">Table of n, a(n) for n = 1..10000</a>

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%e InvPhi(48) = {65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is a term.

%p filter:= n -> isprime(nops(numtheory:-invphi(n))):

%p select(filter, [$1..400]); # _Robert Israel_, Mar 29 2020

%o (PARI) is(k) = isprime(invphiNum(k)); \\ _Amiram Eldar_, Nov 15 2024, using _Max Alekseyev_'s invphi.gp

%Y Cf. A000010, A007366, A007367, A014197, A058277, A060668, A060670, A060674, A063512, A071386-A071389.

%K nonn

%O 1,2

%A _Labos Elemer_, May 23 2002