A171131 - OEIS

%I #20 Aug 01 2024 18:16:04

%S 5,7,19,67,4099,65539,262147,1073741827

%N Primes p such that sum of divisors of p-3 is prime.

%C No further terms up to the 10 millionth prime. - _Harvey P. Dale_, Apr 30 2012

%C If the sum of divisors of a number k is a prime (i.e., k is in A023194), then k is a prime power. If p is prime and p-3 is a prime power, then p-3 is even, so p-3 is a power of 2. Since p-3 = 2^m then sigma(2^m) = 2^(m+1)-1 is a prime. Therefore, all the terms are primes of the form 2^m+3 where m+1 is a Mersenne exponent (A000043), i.e., m is in the intersection of A057732 and {A000043(n)-1}. So, m = 1, 2, 4, 6, 12, 16, 18, 30, and no other value <= A057732(58) = 2205444. Therefore, a(9) > 2^2205444, if it exists. - _Amiram Eldar_, Aug 01 2024

%e 5 is a term since it is a prime and sigma(5-3) = 3 is a prime.

%e 7 is a term since it is a prime and sigma(7-3) = 7 is a prime.

%e 19 is a term since it is a prime and sigma(19-3) = 31 is a prime.

%t f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[PrimeQ[f[p-3]],AppendTo[lst,p]],{n,2*10!}];lst

%t Select[Prime[Range[10000000]],PrimeQ[DivisorSigma[1,#-3]]&] (* _Harvey P. Dale_, Apr 30 2012 *)

%Y Cf. A000043, A000203, A023194, A057732, A171130, A153503.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Dec 04 2009

%E a(8)-a(10) from _Vincenzo Librandi_, Feb 04 2013

%E Two wrong terms removed by _Amiram Eldar_, Aug 01 2024