

A276493


Perfect numbers whose sum of prime factors is prime.


3




OFFSET

1,1


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.


LINKS

Table of n, a(n) for n=1..4.


EXAMPLE

a(1) = (2^21)*2^(21) = 6 because both 2^21 = 3 and 2^2+2*23 = 5 are primes.
a(2) = (2^31)*2^(31) = 28 because both 2^31 = 7 and 2^3+2*33 = 11 are primes.
a(3) = (2^71)*2^(71) = 8128 because both 2^71 = 127 and 2^7+2*73 = 139 are primes.


MAPLE

A276493:=n>`if`(isprime(n) and isprime(2^n1) and isprime(2^n+2*n3), (2^n1)*2^(n1), 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 *)


PROG

(MAGMA) [(2^p1)*2^(p1): p in PrimesUpTo(2000)  IsPrime(2^p+2*p3)];
(MAGMA) [(2^n1)*2^(n1): n in [1..200]  IsPrime(n) and IsPrime(2^n1) and IsPrime(2^n+2*n3)]; // Vincenzo Librandi, Sep 06 2016


CROSSREFS

Subsequence of A000396. Subsequence of A100118.
Cf. A000043, A000668, A007013, A192436, A276511.
Sequence in context: A057246 A154895 A330163 * A074849 A189373 A156927
Adjacent sequences: A276490 A276491 A276492 * A276494 A276495 A276496


KEYWORD

nonn


AUTHOR

JuriStepan Gerasimov, Sep 05 2016


STATUS

approved



