cp's OEIS Frontend

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

%I A294729 #21 Sep 08 2022 08:46:20
%S A294729 6,8,10,14,15,25,26,27,29,34,38,43,47,53,62,65,71,73,79,85,93,122,133,
%T A294729 179,193,211,254,257,277,283,311,331,349,353,389,409,443,467,499,563,
%U A294729 577,599,613,631,643,647,683,709,751,769,829,919,941,1039,1103,1117
%N A294729 Numbers n such that 2^n - 1 is the product of three primes.
%C A294729 The eighteenth composite term is 3481. No other composite terms up to 10000.
%H A294729 Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM"> Integer factorization calculator</a>
%H A294729 John Brillhart et al., <a href="http://www.ams.org/books/conm/022/">Cunningham Project</a> [Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers]
%H A294729 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>
%e A294729 a(1) = 6 because 2^6 - 1 = 63 = 3^2*7 is a 3-almost prime.
%e A294729 a(2) = 8 because 2^8 - 1 = 255 = 3*5*17 is a 3-almost prime.
%t A294729 ParallelMap[ If[ PrimeOmega[2^# - 1] == 3, #, Nothing] &, Range@1250] (* _Robert G. Wilson v_, Nov 28 2017 *)
%o A294729 (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;
%o A294729 (PARI) is(n)=bigomega(2^n-1)==3
%Y A294729 Cf. A000043 (product of one prime), A000225, A085724 (product of two primes), A135977.
%K A294729 nonn
%O A294729 1,1
%A A294729 _Arkadiusz Wesolowski_, Nov 07 2017
%E A294729 a(28)-a(56) added from the Cunningham project