A249759 - OEIS

%I #38 Sep 08 2022 08:46:10

%S 3,5,17,65537

%N Primes p such that sigma(p-1) is a prime q.

%C Subsequence of {A023194(n)+1}.

%C Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.

%C 3 is the only prime p such that sigma(p+1) is prime, i.e., 3 is the only prime p such that sigma(p-1) and sigma(p+1) are both primes.

%C Conjecture: 3 is the only number n such that n and sigma(n+1) are both prime.

%C Primes p such that A051027(p-1) = sigma(sigma(p-1)) = 2*(p-1). Subsequence of A256438. - _Jaroslav Krizek_, Mar 29 2015

%C From _Jaroslav Krizek_, Mar 17 2016: (Start)

%C Primes p such that A000203(A000010(p)) = sigma(phi(p)) is a prime.

%C Prime terms from A062514 and A270413, A270414, A270415 and A270416. (End)

%C From _Jaroslav Krizek_, Nov 27 2016: (Start)

%C Corresponding values of primes q are in A249761: 3, 7, 31, 131071, ...

%C Conjecture: subsequence of A256438 and A278741.

%C Conjecture: also primes p such that tau(p-1) is a prime q; corresponding values of primes q are 2, 3, 5, 17. (End)

%F a(n) = A249760(n) + 1.

%F Sigma(a(n)-1) = A249761(n).

%e Prime 17 is in the sequence because sigma(17-1) = sigma(16) = 31 (prime).

%p with(numtheory): A249759:=n->`if`(isprime(n) and isprime(sigma(n-1)), n, NULL): seq(A249759(n), n=1..6*10^5); # _Wesley Ivan Hurt_, Nov 14 2014

%t Select[Range[10^5], PrimeQ[#]&& PrimeQ[DivisorSigma[1, # - 1]] &] (* _Vincenzo Librandi_, Nov 14 2014 *)

%t Select[Prime[Range[7000]],PrimeQ[DivisorSigma[1,#-1]]&] (* _Harvey P. Dale_, Jun 14 2020 *)

%o (Magma) [p: p in PrimesUpTo(1000000) | IsPrime(SumOfDivisors(p-1))]

%o (PARI) lista(nn) = {forprime(p=1, nn, if (isprime(sigma(p-1)), print1(p, ", ")););} \\ _Michel Marcus_, Nov 14 2014

%Y Cf. A000203, A000668, A019434, A023194, A249760, A249761.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Nov 13 2014