A171131 Primes p such that sum of divisors of p-3 is prime.

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

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 04 2009

Keywords

• nonn

Comments

No further terms up to the 10 millionth prime. - Harvey P. Dale, Apr 30 2012

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

Examples

			5 is a term since it is a prime and sigma(5-3) = 3 is a prime.
7 is a term since it is a prime and sigma(7-3) = 7 is a prime.
19 is a term since it is a prime and sigma(19-3) = 31 is a prime.
		

Crossrefs

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

Programs

Mathematica

f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[PrimeQ[f[p-3]],AppendTo[lst,p]],{n,2*10!}];lst
Select[Prime[Range[10000000]],PrimeQ[DivisorSigma[1,#-3]]&] (* Harvey P. Dale, Apr 30 2012 *)

Extensions

a(8)-a(10) from Vincenzo Librandi, Feb 04 2013

Two wrong terms removed by Amiram Eldar, Aug 01 2024