%I #20 Jan 23 2022 20:04:29
%S 2,4,16,65536
%N Numbers k such that k+1 and sigma(kn) are both primes.
%C 4 is the only number k such that k-1 and sigma(k) are both primes.
%C Corresponding values of k+1 and sigma(k) are in A249759 and A249761.
%C Conjectures: (1) sequence is finite; (2) a(n) + 1 is a Fermat prime (A019434); (3) sigma(a(n)) is a Mersenne prime (A000668).
%C Subsequence of A023194, and by a comment in that entry it follows that each term is a prime power. From that conjectures (2) and (3) above easily follow. - _Jeppe Stig Nielsen_, Jan 13 2015
%F a(n) = A249759(n) - 1.
%e 16 is a term because 16+1=17 and sigma(16)=31 are both primes.
%t Select[Range[10^5], PrimeQ[# + 1]&& PrimeQ[DivisorSigma[1, #]] &] (* _Vincenzo Librandi_, Nov 14 2014 *)
%o (Magma) [n: n in [1..10^8] | IsPrime(n+1) and IsPrime(SumOfDivisors(n))]
%Y Cf. A000203, A000668, A019434, A249759, A249761.
%K nonn,more
%O 1,1
%A _Jaroslav Krizek_, Nov 13 2014