A294729 Numbers n such that 2^n - 1 is the product of three primes.

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

Author

Arkadiusz Wesolowski, Nov 07 2017

Keywords

• nonn

Comments

The eighteenth composite term is 3481. No other composite terms up to 10000.

Examples

			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.
		

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

Crossrefs

Cf. A000043 (product of one prime), A000225, A085724 (product of two primes), A135977.

Programs

Magma

lst:=[]; factors:=func; bigomega:=func; IsCube:=func; 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;

Mathematica

ParallelMap[ If[ PrimeOmega[2^# - 1] == 3, #, Nothing] &, Range@1250] (* Robert G. Wilson v, Nov 28 2017 *)

PARI

is(n)=bigomega(2^n-1)==3

Extensions

a(28)-a(56) added from the Cunningham project