OFFSET
1,2
COMMENTS
tau(m) is the number of divisors of m (A000005).
Numbers m such that A046798(m) = m + 1.
Corresponding values of tau(2^m): 2, 64, 512, ...
Corresponding pairs of numbers (2^m, 2^m + 1): (2, 3); (9223372036854775808, 9223372036854775809); ...
All terms in the sequence are odd. (If m were an even number m = 2k, then tau(2^m) = m + 1 = 2k + 1 would be odd, but tau(2^m + 1) = tau(2^(2k) + 1) = tau(4^k + 1) would be even, since 4^k + 1 can never be a square.) - Jon E. Schoenfield, Mar 26 2017
Each of the known terms a(1), a(2), and a(3) is of the form 2^j - 1 (so tau(2^m) = 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 40-digit prime times a 537-digit composite (coprime to each of the known 7 prime factors), 2047 will be a term iff that 537-digit composite has exactly 2048/2^7 = 16 divisors. - Jon E. Schoenfield, Mar 30 2017; updated by Max Alekseyev, Feb 16 2023
Candidate next terms: 1151?, 1187?, 1231?, 1259?, 1319?, 1343?, 1399?, 1471?, 1535?, 1567?, 1583?, 1663?, 1759?, 1847?, 1913?, 1919?, 2047?. In particular, a(4) >= 1151. - Max Alekseyev, Feb 16 2023
EXAMPLE
For m = 1, tau(2) = tau(3) = 2.
PROG
(Magma) [n: n in [0..500] | NumberOfDivisors(2^n) eq NumberOfDivisors(2^n + 1)]
CROSSREFS
KEYWORD
nonn,more,bref
AUTHOR
Jaroslav Krizek, Mar 18 2017
STATUS
approved