A153503 -id:A153503 - cp's OEIS frontend

A229065 Numbers of the form 2^(p-1)+3, where p is prime.

5, 7, 19, 67, 1027, 4099, 65539, 262147, 4194307, 268435459, 1073741827, 68719476739, 1099511627779, 4398046511107, 70368744177667, 4503599627370499, 288230376151711747, 1152921504606846979, 73786976294838206467, 1180591620717411303427, 4722366482869645213699

Offset: 1

Vincenzo Librandi, Sep 17 2013

nonn

Primes in the sequence: 5, 7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, ...

On the other hand, for example, 2^(p-1) + 3 is composite when p == 11 (mod 12) or p == 5 (mod 18), with p>5; or when p is of the form 2*h^2+2*h*(k+2)+3*k, with k>0 and h>1.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A153503 (associated primes p), A098828, A057732, A057736.

Magma
```
[2^(p-1)+3:  p in PrimesUpTo(80)];
```
Mathematica
```
Table[2^(Prime[n] - 1) + 3, {n, 25}]
```

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

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 04 2009

nonn

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

			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.

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

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 *)

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

Two wrong terms removed by Amiram Eldar, Aug 01 2024

cp's OEIS Frontend

A229065 Numbers of the form 2^(p-1)+3, where p is prime.

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