A357916 - OEIS

%I #14 Feb 29 2024 13:45:32

%S 2,3,5,13,23,59,113,137,229,457,509,523,661,1021,2063,3541,3923,4973,

%T 5449,5521,9949,10103,10273,12659,14107,15601,16249,17033,22063,25321,

%U 29759,32507,34843,36293,37273,52501,54059,62753,68449,68909,89329,99409,103963,111347,125509,139297,146309,157231

%N Primes p that can be written as phi(k) + d(k) for some k, where phi(k) = A000010(k) is Euler's totient function and d(k) = A000005(k) is the number of divisors of k.

%C Does any prime have more than one representation as phi(k) + d(k)?

%H Robert Israel, <a href="/A357916/b357916.txt">Table of n, a(n) for n = 1..3000</a>

%e a(4) = 13 is a term because 13 is prime and for k = 16, phi(k) + d(k) = 8 + 5 = 13.

%p N:= 10^6: # to allow k <= N

%p pmax:=  evalf(N/(exp(gamma)*log(log(N))+3/log(log(N)))): # lower bound for phi(k), k<=N

%p P:= {3}:

%p for k from 1 to sqrt(N) do

%p   n:= k^2;

%p   v:= numtheory:-phi(n)+numtheory:-tau(n);

%p   if v <= pmax and isprime(v) then

%p      P:= P union {v};

%p   fi

%p od:

%p sort(convert(P,list));

%t Select[Table[EulerPhi[n]+DivisorSigma[0,n],{n,400000}],PrimeQ]//Sort (* _Harvey P. Dale_, Feb 29 2024 *)

%Y Cf. A000005, A000010, A061468, A357917.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Oct 19 2022