A253849 - OEIS

%I #20 Sep 08 2022 08:46:11

%S 2,4,9,16,25,64

%N Numbers n such that 2^sigma(n) - 1 is a prime.

%C Also numbers n such that sigma(n) is in A000043, i.e., p = 2^sigma(n) - 1 is a Mersenne prime (A000668). The sequence of corresponding primes p reads: 7, 127, 8191, 2147483647, 2147483647, 170141183460469231731687303715884105727, ..., see A253851.

%C Subsequence of A023194 (numbers n such that sigma(n) is a prime), see there for an explanation why all terms except the first one are squares.

%C The sequence of values of sigma(a(n)) is 3, 7, 13, 31, 31, 127, ... and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043). See A253850.

%C Sequence differs from A023194 because A023194(7) = 289 but if a(7) exists, it must be a number n such that sigma(n) > A000043(43) = 30402457.

%C a(n) must be an even power of a prime. If it is the square of an odd prime, then this prime must be in A053182. If a(n) is an even power of 2, a(n)=2^(2k), then sigma(a(n))=2^(2k+1)-1. Thus, 2k+1 must be a double Mersenne prime exponent, i.e., such that the corresponding Mersenne prime is again a Mersenne exponent, cf. A103901. Only 4 such primes are known, and a(6)=2^6 (k=3) corresponds to the largest known prime of this type, 2^(2k+1)-1 = 127. - _M. F. Hasler_, Jan 21 2015

%e 4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.

%t a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* _Michael De Vlieger_, Jan 19 2015 *)

%o (Magma) [n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)]

%Y Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Jan 16 2015