A357916 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.

2, 3, 5, 13, 23, 59, 113, 137, 229, 457, 509, 523, 661, 1021, 2063, 3541, 3923, 4973, 5449, 5521, 9949, 10103, 10273, 12659, 14107, 15601, 16249, 17033, 22063, 25321, 29759, 32507, 34843, 36293, 37273, 52501, 54059, 62753, 68449, 68909, 89329, 99409, 103963, 111347, 125509, 139297, 146309, 157231

Offset

1,1

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..3000

Example

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

Maple

N:= 10^6: # to allow k <= N
pmax:=  evalf(N/(exp(gamma)*log(log(N))+3/log(log(N)))): # lower bound for phi(k), k<=N
P:= {3}:
for k from 1 to sqrt(N) do
  n:= k^2;
  v:= numtheory:-phi(n)+numtheory:-tau(n);
  if v <= pmax and isprime(v) then
     P:= P union {v};
  fi
od:
sort(convert(P, list));

Mathematica

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

Crossrefs

Cf. A000005, A000010, A061468, A357917.

Sequence in context: A282236 A215306 A072537 * A289758 A089477 A099962

Adjacent sequences: A357913 A357914 A357915 * A357917 A357918 A357919

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Oct 19 2022

Status

approved