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%I #11 Mar 25 2023 05:16:48
%S 1,1,2,1,6,1,18,2,2,3,8,1,40,9,2,1,177728,1,120,3,6,4,32906,95,868,20,
%T 1648,346,1996040,1,700,623,134,88864,284,1236,821688,60
%N Smallest k such that sigma(2^(k*n)) is prime.
%C Equivalently, smallest k such that k*n + 1 is a Mersenne exponent (A000043).
%C As of Mar 11 2017, the j-th Mersenne exponent A000043(j) is known for j=1..45; four additional terms of A000043 are listed in the Extensions for that sequence, but it is not yet known whether they are A000043(46) through A000043(49). None of the first 45 Mersenne exponents are of the form k*29 + 1, so a(29) > floor((A000043(45) - 1)/29) = floor((37156667 - 1)/29) = 1281264. However, one of the four additional terms is 57885161 = 1996040*29 + 1; thus, 1281264 < a(29) <= 1996040.
%C a(39) > floor((A000043(45) - 1)/39) = 952735.
%C This sequence coincides with A186283 (Least number k such that k*n+1 is a prime dividing 2^n-1) from a(2) through a(8), but a(9) = 2 whereas A186283(9) = 8.
%e a(1) = 1 because sigma(2^(1*1)) = sigma(2) = 1 + 2 = 3 is prime. (1*1 + 1 = 2 = A000043(1).)
%e a(3) = 2 because sigma(2^(1*3)) = sigma(2^3) = 1 + 2 + 4 + 8 = 15 is not prime, but sigma(2^(2*3)) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 is prime. (1*3 + 1 = 4 is not in A000043, but 2*3 + 1 = 7 = A000043(4).)
%e a(17) = 177728 because sigma(2^(177728*17)) is prime and sigma(2^(k*17)) is not prime for any k < 177728. (177728*17 + 1 = 3021377 = A000043(37), and no Mersenne exponent less than A000043(37) is of the form k*17 + 1.)
%o (PARI) a(n) = k=1; while(!isprime(sigma(2^(k*n))), k++); k; \\ _Michel Marcus_, Mar 12 2017
%Y Cf. A000043, A000203, A186283, A279004.
%K nonn,hard,more
%O 1,3
%A _Jon E. Schoenfield_, Mar 11 2017
%E a(29)-a(38) from _Jinyuan Wang_, Mar 25 2023
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