login
A294729
Numbers n such that 2^n - 1 is the product of three primes.
0
6, 8, 10, 14, 15, 25, 26, 27, 29, 34, 38, 43, 47, 53, 62, 65, 71, 73, 79, 85, 93, 122, 133, 179, 193, 211, 254, 257, 277, 283, 311, 331, 349, 353, 389, 409, 443, 467, 499, 563, 577, 599, 613, 631, 643, 647, 683, 709, 751, 769, 829, 919, 941, 1039, 1103, 1117
OFFSET
1,1
COMMENTS
The eighteenth composite term is 3481. No other composite terms up to 10000.
LINKS
Dario Alejandro Alpern, Integer factorization calculator
John Brillhart et al., Cunningham Project [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
S. S. Wagstaff, Jr., The Cunningham Project
EXAMPLE
a(1) = 6 because 2^6 - 1 = 63 = 3^2*7 is a 3-almost prime.
a(2) = 8 because 2^8 - 1 = 255 = 3*5*17 is a 3-almost prime.
MATHEMATICA
ParallelMap[ If[ PrimeOmega[2^# - 1] == 3, #, Nothing] &, Range@1250] (* Robert G. Wilson v, Nov 28 2017 *)
PROG
(Magma) lst:=[]; factors:=func<n | Factorization(n)>; bigomega:=func<n | &+[d[2]: d in factors(n)]>; IsCube:=func<n | (n^(1/3)) eq Floor(n^(1/3))>; for n in [2..254] do if IsPrime(n) then if bigomega(2^n-1) eq 3 then Append(~lst, n); end if; else f:=factors(n); a:=f[1, 1]; if IsPrime(2^a-1) then if IsSquarefree(n) then if bigomega(n) eq 2 then b:=f[2, 1]; if IsPrime(2^b-1) and IsPrime(Truncate((2^n-1)/((2^a-1)*(2^b-1)))) then Append(~lst, n); end if; end if; end if; if IsSquare(n) or IsCube(n) then if bigomega(Truncate((2^n-1)/(2^a-1))) eq 2 then Append(~lst, n); end if; end if; end if; end if; end for; lst;
(PARI) is(n)=bigomega(2^n-1)==3
CROSSREFS
Cf. A000043 (product of one prime), A000225, A085724 (product of two primes), A135977.
Sequence in context: A007422 A030513 A161918 * A242270 A298252 A152126
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(28)-a(56) added from the Cunningham project
STATUS
approved