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a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.
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%I #16 Jan 28 2018 18:41:40

%S 1,0,3,7,0,31,0,127,217,381,889,0,3937,8191,11811,27559,57337,131071,

%T 253921,524287,1040257,1777447,4063201,7281799,16646017,32247967,

%U 66584449,116522119,225735769,516026527,1073602561,2147483647,4294434817,7515217927,15032385529

%N a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

%C If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

%F a(A078426(n)) = 0.

%F a(A180221(n)) > 0.

%F a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - _Michael B. Porter_, Nov 14 2017

%e a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0.

%e a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5.

%e a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.

%o (PARI) a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ _Michael B. Porter_, Nov 14 2017

%o (PARI) a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ _Rémy Sigrist_, Jan 08 2018

%Y Cf. A000043, A000203, A046528, A048947, A057637, A180221.

%Y Cf. A247956 (the smallest number k instead of the largest).

%Y Cf. A078426 (no solution to the equation sigma(x)=2^n).

%Y A000668 (Mersenne primes) is a subsequence.

%K nonn

%O 0,3

%A _Jaroslav Krizek_, Nov 13 2017