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

1,1

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:

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

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

Links

Table of n, a(n) for n=1..11.

Jean-Luc Garambois, Aliquot sequences starting on integer powers n^i.

Mersenne forum, Results presentation page.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A083500 A102305 A160842 * A007745 A188556 A021011

Adjacent sequences: A365348 A365349 A365350 * A365352 A365353 A365354

Keyword

nonn,hard,more

Author

Jean Luc Garambois, Sep 02 2023

Status

approved