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%I #17 Jan 16 2023 21:41:16
%S 3,5,17,65,1025,4097,65537,262145,4194305,268435457,1073741825,
%T 68719476737,1099511627777,4398046511105,70368744177665,
%U 4503599627370497,288230376151711745,1152921504606846977,73786976294838206465,1180591620717411303425,4722366482869645213697
%N Odd numbers k such that tau(k-1) is a prime.
%C tau(k) = A000005(k) = the number of divisors of k.
%C Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...
%C Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.
%F a(n) = A061286(n) + 1.
%F sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.
%F tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).
%e Odd number 65 is in the sequence because tau(64) = 7 (prime).
%o (Magma) [n: n in[2..10000000] | IsOdd(n) and IsPrime(NumberOfDivisors(n-1))]
%o (PARI) isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ _Michel Marcus_, Nov 27 2016
%Y Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Nov 27 2016