A309981 a(n) is the smallest number k such that the value of n can be deduced given only the values tau(n), tau(n+1), ..., tau(n+k), where tau is the number of divisors function.
0, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 3, 2, 1, 1, 3, 2, 4, 4, 3, 2, 2, 1, 4, 4, 4, 4, 4, 4, 5, 4, 3, 2, 1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 1, 2, 3, 6, 6, 6, 6, 5, 4, 4, 4, 5, 4, 3, 2, 1, 1, 6, 6, 5, 5, 6, 5, 5, 4, 6, 6, 6, 5, 4, 3, 2, 1, 2, 6, 6, 6, 5, 4, 6
Offset: 1
Keywords
Examples
a(1) = 0 is the only occurrence of 0 since n = 1 is the only number that has only 1 divisor and therefore tau(n) = 1 uniquely identifies n = 1. For any other number we need more tau values (tau(n+j), 0 <= j <= k) in order to identify n through these k+1 values. a(2) = 1: n = 2 is the only prime such that n+1 is also prime, therefore tau(n) = 2 and tau(n+1) = 2 implies n = 2 (and only tau(n) = 2 is not enough). a(3) = 1: n = 3 is the only prime such that n+1 is the square of a prime, i.e., n+1 has 3 divisors, so tau(n) = 2 and tau(n+1) = 3 uniquely identify n = 3. a(4) = 1: n = 4 is the only square of a prime such that n+1 is a prime. (For all other n = p^2 with p prime, n+1 is an even composite.) a(5) = 2: the numbers tau(n) and tau(n+1) are 2 and 4, respectively, for n = 5, 7, 13, 37, 61, ..., but 5 is the only number n such that (tau(n), tau(n+1), tau(n+2)) = (2, 4, 2). (Indeed this means that n, n+2 are twin primes, separated by a semiprime n+1. But all larger twin primes are of the form 6m+-1, so the intermediate number is a larger multiple of 6 with more than 4 divisors.) From _M. F. Hasler_, Mar 31 2023: (Start) a(6) = 2: Here tau(n) = 4, tau(n+1) = 2, tau(n+2) = 4 mean that n is a semiprime or p^3, followed by a prime and then another semiprime or q^3. For larger n, prime powers are excluded through parity ((n, n+1) = (p^3, p') and/or (n+1, n+2) = (p', q^3) are all odd), and one can't have two even semiprimes separated by 2, n = 2p, n+2 = 2q. So n = 6 is the only such number. (End) a(49) = 2: (tau(n), tau(n+1)) = (3,6) for n = 49, 1681, and only two other known values (i.e., for squares of terms > 3 in A086397), but (tau(n), tau(n+1), tau(n+2)) = (3, 6, 4) only for n = 49 (which is the only square of a prime p such that both sqrt((p^2 + 1)/2) and (p^2 + 2)/3 are also prime).
Links
- Jon E. Schoenfield, Table of n, m(n) for n = 1..87, where m(n) is the smallest nonnegative integer (other than n) such that tau(m+j) = tau(n+j) for all j in 0..a(n)-1 (corrected Apr 20 2023, with thanks to M. F. Hasler for pointing out that there were errors in the original file).
Formula
a(1)=0; for n > 1, a(n) = 1 iff n is a term of A161460.
Comments