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

%I A365351 #39 Aug 17 2025 02:28:00
%S A365351 6,11,18,27,41,74,157,197,294,549,581
%N A365351 Exponents e such that the aliquot sequence starting with 2^e ends with a prime number at index 2.
%C A365351 That is, exponents e such that s(s(2^e)) is prime, where s(n) = sigma(n)-n (A001065).
%C A365351 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).
%C A365351 From _Amiram Eldar_, Sep 02 2023: (Start)
%C A365351 Numbers k such that 2^k - 1 is a term of A037020.
%C A365351 1206 < a(12) <= 2351 (2351 is a term). (End)
%H A365351 Jean-Luc Garambois, <a href="http://www.aliquotes.com/aliquotes_puissances_entieres/aliquotes_puissances_entieres.html">Aliquot sequences starting on integer powers n^i</a>.
%H A365351 Mersenne forum, <a href="https://www.mersenneforum.org/showpost.php?p=637222&amp;postcount=2427">Results presentation page</a>.
%t A365351 Select[Range[100], PrimeQ[DivisorSigma[1, 2^# - 1] - 2^# + 1] &] (* _Amiram Eldar_, Sep 02 2023 *)
%o A365351 (Sage)
%o A365351 def s(n):
%o A365351     sn = sigma(n) - n
%o A365351     return sn
%o A365351 e = 1
%o A365351 exponents_list = []
%o A365351 while e<=200:
%o A365351     m = 2^e
%o A365351     index = 0
%o A365351     if is_prime(s(s(m))):
%o A365351         exponents_list.append(e)
%o A365351     e+=1
%o A365351 print (exponents_list)
%o A365351 (PARI) f(n) = sigma(n) - n; \\ A001065
%o A365351 isok(k) = ispseudoprime(f(f(2^k))); \\ _Michel Marcus_, Sep 02 2023
%Y A365351 Cf. A000043 (Mersenne exponents), A001065, A037020.
%K A365351 nonn,hard,more
%O A365351 1,1
%A A365351 _Jean Luc Garambois_, Sep 02 2023