OFFSET
2,1
COMMENTS
Also the number of zerofree numbers <= A324155(n).
Expressed in base n - 1 and starting with n = 3, the sequence is 1011110, 11111111110, 1111111111111110, 411111111111111111110, 211111111111111111111111110, 211111111111111111111111111111110, 211111111111111111111111111111111111110, 1111111111111111111111111111111111111111111110, 1111111111111111111111111111111111111111111111111110, ....
Ostensibly, the reason for that is the calculation formula (see Formula section) for the number of zerofree numbers <= x^m + y, with y < (x^(m+1)-1)/(x-1) - x^m. But the deeper reason is the definition of sequence A324155. Each term A324155(n) marks a point of intersection between the curve numOfZerofreeNum_n(x) [the number of base-n zerofree numbers <= x] and the curve pi(x) [the number of prime numbers <= x]. Since numOfZerofreeNum_n(x) doesn't change for relatively large intervals at x = k*n^m (approx. a portion of > 1/(k*n)), but grows similar to pi(x) for regions outside, it is likely, that the point of intersection lies between x = k*n^m and x = n^m*(k + 1/n + 1/n^2 + 1/n^3 + ... + 1/n^m). The chance is maximal for k = 1, since the density of primes becomes smaller for greater x. Nevertheless, k > could also happen as we can see for n = 6, 7, 8 and 9.
FORMULA
a(n) = pi(A324155(n)).
a(n) = numOfZerofreeNum_n(A324155(n)), where numOfZerofreeNum_n(x) is the number of base-n zerofree numbers <= x (cf. A324161).
a(n) = k*(n-1)^m + ((n-1)^m - 1)/(n-2) - 1, where m = floor(log_n(A324155(n))), k = floor(A324155(n)/n^m), and provided A324155(n) - k*n^m < (n^(m+1)-1)/(n-1) - n^m.
With d := log(n-1)/log(n):
a(n) <= ((n - 1)*(A324155(n) + 1)^d - 1)/(n - 2) - 1.
a(n) >= (((n - 1)*A324155(n) + n)^d - 1)/(n - 2) - 1.
EXAMPLE
CROSSREFS
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Mar 05 2019
STATUS
approved