A276493 Perfect numbers whose sum of prime factors is prime.

6, 28, 8128, 14474011154664524427946373126085988481573677491474835889066354349131199152128

Offset: 1

Author

Juri-Stepan Gerasimov, Sep 05 2016

Keywords

• nonn

Comments

The next term is too large to include.

Numbers (2^n - 1)*2^(n - 1) such that both 2^n - 1 and 2^n + 2*n - 3 are prime.

Conjectures (defining x = 170141183460469231731687303715884105727 = A007013(4)):

(1) (2^x - 1)*2^(x - 1) is a term because 2^x - 1 and 2^x + 2*x - 3 are primes;

(2) a(n) is equal to (2^A007013(k) - 1)*2^(A007013(k) - 1) such that 2^A007013(k) - 1 and 2^A007013(k) + 2*A007013(k) - 3 are primes for some prime value of A007013(k) where k => 0;

(3) primes of A007013 are Mersenne prime exponents A000043, i.e. x is new exponent in A000043.

Examples

			a(1) = (2^2-1)*2^(2-1) = 6 because both 2^2-1 = 3 and 2^2+2*2-3 = 5 are primes.
a(2) = (2^3-1)*2^(3-1) = 28 because both 2^3-1 = 7 and 2^3+2*3-3 = 11 are primes.
a(3) = (2^7-1)*2^(7-1) = 8128 because both 2^7-1 = 127 and 2^7+2*7-3 = 139 are primes.
		

Crossrefs

Subsequence of A000396. Subsequence of A100118.

Cf. A000043, A000668, A007013, A192436, A276511.

Programs

Magma

[(2^p-1)*2^(p-1): p in PrimesUpTo(2000) | IsPrime(2^p+2*p-3)];

Magma

[(2^n-1)*2^(n-1): n in [1..200] | IsPrime(n) and IsPrime(2^n-1) and IsPrime(2^n+2*n-3)]; // Vincenzo Librandi, Sep 06 2016

Maple

A276493:=n->`if`(isprime(n) and isprime(2^n-1) and isprime(2^n+2*n-3), (2^n-1)*2^(n-1), NULL): seq(A276493(n), n=1..10^3); # Wesley Ivan Hurt, Sep 07 2016

Mathematica

Select[PerfectNumber[Range[12]],PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[#]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 06 2020 *)