A180646 Numbers k such that 3 + phi(k)^3 is prime, where phi is Euler's totient function.

3, 4, 5, 6, 8, 10, 12, 17, 23, 32, 34, 40, 46, 48, 51, 53, 60, 64, 68, 80, 85, 96, 101, 102, 106, 107, 120, 125, 128, 136, 159, 160, 167, 170, 191, 192, 202, 204, 212, 213, 214, 235, 240, 249, 250, 267, 281, 284, 318, 319, 321, 332, 334, 339, 345, 355, 356, 368

Offset

1,1

Comments

The sequence appears to be infinite, but I have no proof. There are many consecutive values: e.g., 3,4,5,6; 101,102; 212,213,214; 355,356.

The generalized Bunyakovsky conjecture implies that there are infinitely many primes in the sequence, i.e. primes p such that (p-1)^3+3 = phi(p)^3+3 is prime (sequence A333199). - Robert Israel, Mar 11 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(5) = 102 since 3 + phi(102)^3 = 3 + 1061208 = 1061211 is prime.

Maple

select(t -> isprime(numtheory:-phi(t)^3+3), [$1..1000]); # Robert Israel, Mar 11 2020

Mathematica

Select[Range[1000], PrimeQ[3 + EulerPhi[#]^3] &] (* Indranil Ghosh, Apr 02 2017 *)

Prog

(PARI) isok(n) = isprime(3+eulerphi(n)^3); \\ Michel Marcus, Apr 02 2017
(Python)
from sympy import isprime, totient
print([n for n in range(1001) if isprime(3 + totient(n)**3)]) # Indranil Ghosh, Apr 02 2017

Crossrefs

Cf. A000010, A333199.

Sequence in context: A130216 A120162 A002859 * A300217 A062514 A065875

Adjacent sequences: A180643 A180644 A180645 * A180647 A180648 A180649

Keyword

nonn

Author

Carmine Suriano, Sep 14 2010

Status

approved