A365351 Exponents e such that the aliquot sequence starting with 2^e ends with a prime number at index 2.

6, 11, 18, 27, 41, 74, 157, 197, 294, 549, 581

Offset: 1

Author

Jean Luc Garambois, Sep 02 2023

Keywords

• nonn
• hard
• more

Comments

That is, exponents e such that s(s(2^e)) is prime, where s(n) = sigma(n)-n (A001065).

Note that exponents e such that aliquot sequences starting with 2^e end with a prime number at index 1 (exponents e such that s(2^e) is prime) are called "Mersenne exponents" (see A000043).

From Amiram Eldar, Sep 02 2023: (Start)

Numbers k such that 2^k - 1 is a term of A037020.

1206 < a(12) <= 2351 (2351 is a term). (End)

Links

• Jean-Luc Garambois, Aliquot sequences starting on integer powers n^i.
• Mersenne forum, Results presentation page.

Crossrefs

Cf. A000043 (Mersenne exponents), A001065, A037020.

Programs

Mathematica

Select[Range[100], PrimeQ[DivisorSigma[1, 2^# - 1] - 2^# + 1] &] (* Amiram Eldar, Sep 02 2023 *)

PARI

f(n) = sigma(n) - n; \\ A001065
isok(k) = ispseudoprime(f(f(2^k))); \\ Michel Marcus, Sep 02 2023

Sage

def s(n):
    sn = sigma(n) - n
    return sn
e = 1
exponents_list = []
while e<=200:
    m = 2^e
    index = 0
    if is_prime(s(s(m))):
        exponents_list.append(e)
    e+=1
print (exponents_list)