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A253849 Numbers n such that 2^sigma(n) - 1 is a prime. 3
2, 4, 9, 16, 25, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
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.
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.
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.
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
LINKS
EXAMPLE
4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.
MATHEMATICA
a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* Michael De Vlieger, Jan 19 2015 *)
PROG
(Magma) [n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)]
CROSSREFS
Sequence in context: A073858 A006474 A110878 * A364131 A077137 A023194
KEYWORD
nonn,more
AUTHOR
Jaroslav Krizek, Jan 16 2015
STATUS
approved

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Last modified June 20 05:29 EDT 2024. Contains 373512 sequences. (Running on oeis4.)