cp's OEIS Frontend

Examples

			For n=2 and n=3, the first and second differences are coprime for all m. Each of their sequences {b(m)} consist only of 1's, which can be described trivially as [1] with a period of 1, so a(2) = a(3) = 1.
For n > 3, the first and second differences are coprime for some m values, but not for all. Each repeating periodic sequence {b(m)} begins at m=1, and can be used to predict what b(m) will be at any higher m value for that power n.
n=4 has the 5-term repeating sequence, beginning at m=1:
  [0 0 1 1 1], so a(4) = 5.
The sequence is repeating, so for example, f(41)..f(45) is also [0 0 1 1 1].
n=5 has the 11-term repeating sequence
  [1 1 0 1 1 0 1 1 1 1 1]
so a(5) = 11.
n=6 has the 91-term repeating sequence
  [0 0 0 0 0 0 1 0 0 0 0 1 1
   1 0 0 0 0 0 1 1 0 0 0 0 1
   1 1 0 0 0 0 1 1 1 0 0 0 0
   1 1 1 0 0 0 0 1 1 1 0 0 0
   0 1 1 1 0 0 0 0 1 1 1 0 0
   0 0 1 1 0 0 0 0 0 1 1 1 0
   0 0 0 1 0 0 0 0 0 0 1 1 1]
so a(6) = 91.
The period for higher n values has yet to be found. If they exist, it seems they would be quite large given the large expansion from 5, 11, to 91.
Example: the 233rd term in the sequence of values for n=6 is calculated by using m=233 and n=6. Define the first difference for the 233rd term as 234^6 - 233^6 = 4164782373647. The second difference for the 233rd term is 235^6 - 2*234^6 + 233^6 = 89948228762. The terms 4164782373647 and 89948228762 share a common factor, so the 233rd term of the sequence for 6th powered terms is denoted 0 (not coprime). Because the 6th powered terms repeat their tendency of being coprime or not every 91 terms, we could instead look at 233 mod 91 = 51, and from the table for n=6 above, the 51st term is 0.