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A215896 Largest k = 2^(m - 1)*(2^m - 1) such that bigomega(k) = n or 0 if no such k exists.

%I A215896 #20 Jul 09 2025 04:34:47
%S A215896 1,0,6,28,0,496,0,8128,2016,0,130816,0,2096128,33550336,0,0,134209536,
%T A215896 8589869056,0,137438691328,0,0,0,34359607296,35184367894528,
%U A215896 8796090925056,0,562949936644096,2251799780130816,9007199187632128,140737479966720,2305843008139952128,0
%N A215896 Largest k = 2^(m - 1)*(2^m - 1) such that bigomega(k) = n or 0 if no such k exists.
%C A215896 Largest k = 2^(m-1)*(2^m-1) such that bigomega(k) = prime(n) or 0 if no such k exists (other version): 6, 28, 496, 8128, 0, 0, 8589869056, 137438691328, 34359607296, 9007199187632128, 2305843008139952128, 0, ...
%C A215896 Mersenne exponents (A000043): numbers n such that omega(2^(n-1)*(2^n-1)) = 2, or bigomega(2^(n-1)*(2^n-1)) = n, or tau(2^(n-1)*(2^n-1)) = 2n, or sigma(2^(n-1)*(2^n-1)) = 2^n*(2^n-1).
%C A215896 Smallest k = 2^(m-1)*(2^m-1) such that bigomega(k) = n or 0 if no such k exists : 1, 0, 6, 28, 0, 120, 0, 8128, 2016, 0, 32640, 0, 523776, 33550336, 0, 0, 8386560, 536854528, 0, 2147450880, 0, 0, 0, 34359607296, 2199022206976, 549755289600, 0, 562949936644096, 2251799780130816,...
%e A215896 a(0) = 1 because 2^(1-1)*(2^1-1) = 1 and A001222(1) = 0,
%e A215896 a(2) = 6 because 2^(2-1)*(2^2-1) = 6 and A001222(6) = 2,
%e A215896 a(3) = 28 because 2^(3-1)*(2^3-1) = 28 and A001222(28) = 3,
%e A215896 a(5) = 496 because 2^(4-1)*(2^4-1) = 120, 2^(5-1)*(2^5-1) = 496 and A001222(120) = A001222(496) = 5, 496 > 120.
%e A215896 a(7) = 8128 because 2^(7-1)*(2^7-1) = 8128 and A001222(8128) = 7,
%e A215896 a(8) = 2016 because 2^(6-1)*(2^6-1) = 2016 and A001222(2016) = 8,
%e A215896 a(10) = 130816 because 2^(8-1)*(2^8-1) = 32640, 2^(9-1)*(2^9-1) = 130816 and A001222(32640) = A001222(130816) = 10, 130816 > 32640.
%p A215896 A215896 := proc(n)
%p A215896       local m,k;
%p A215896       for m from n+2 by -1 do
%p A215896         k := 2^(m-1)*(2^m-1) ;
%p A215896         if k < 0 then
%p A215896             return 0 ;
%p A215896         end if;
%p A215896         if numtheory[bigomega](k) = n then
%p A215896             return k ;
%p A215896         end if;
%p A215896     end do:
%p A215896 end proc: # _R. J. Mathar_, Sep 11 2012
%Y A215896 Cf. A000396, A006516, A046051, A144858.
%K A215896 nonn
%O A215896 1,3
%A A215896 _Gerasimov Sergey_, Aug 25 2012