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
Keywords
Examples
a(4) = 13 is a term because 13 is prime and for k = 16, phi(k) + d(k) = 8 + 5 = 13.
Links
- Robert Israel, Table of n, a(n) for n = 1..3000
Programs
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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 *)
Comments