tau(n) is the number of divisors of n (A000005).
There are no other terms <= 880.
Numbers n such that A046801(n) = n + 1.
Numbers n such that A000005(A000079(n)) = A000005(A000051(n)).
Corresponding values of tau(2^n): 2, 64, 512, ...
Corresponding pairs of numbers (2^n, 2^n + 1): (2, 3); (9223372036854775808, 9223372036854775809); ...
From Jon E. Schoenfield, Mar 26 2017 [expanded Mar 30 2017]: (Start)
All terms in the sequence are odd. (If n were an even number n = 2k, then tau(2^n) = n + 1 = 2k + 1 would be odd, but tau(2^n + 1) = tau(2^(2k) + 1) = tau(4^k + 1) would be even, since 4^k + 1 can never be a square.)
2^881 + 1 may be difficult to factor completely, but it is divisible by each of the two primes 22907 and 58147, yet it is divisible by neither 22907^2 nor 58147^2, so each of those two primes will appear in the factorization of 2^881 + 1 with multiplicity 1, and thus tau(2^881 + 1) is some multiple of 4, whereas tau(2^881) = 882 = 2*441 is not, so 881 is not a term. Using partial factorizations of 2^n + 1 for values of n > 881, it is not difficult to show that there are no more terms < 1087.
Each of the known terms a(1), a(2), and a(3) is of the form 2^j  1 (so tau(2^n) = 2^j); the corresponding values of j are 1, 6, and 9. Are there additional terms of this form? The sequence does not include 2^10  1 = 1023; even without completely factoring 2^1023 + 1, it can be seen that 3 must appear in that factorization with a multiplicity of 2, so tau(2^1023 + 1) is divisible by 3, and thus cannot be 1024. Since 2^2047 + 1 is 3 * 179 * 2796203 * 53484017 * 62020897 * 18584774046020617 times a 576digit composite (coprime to each of those 6 prime factors listed), 2047 will be a term iff that 576digit composite has exactly 2048/2^6 = 32 divisors.
(End)
