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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.
3
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
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
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Oct 19 2022
STATUS
approved