cp's OEIS Frontend

In A082905, all binomial coefficients C(n,j) are replaced by C(n/g, j/g), where g=gcd(n,j); a(n) = Sum_{j=0..n-1} C(n/g, j/g).

Compared to A082906, this sequence shows better the drop from 2^n upon replacing every binomial(n,m) in the Newton's expansion of (1+1)^n by the 'reduced' binomial(n/gcd(n,m), m/gcd(n,m)). For n > 1, a(n) is zero if and only if n is prime (no reduction, no drop). The ratio r(n) = a(n)/2^n is always smaller than 1 and presents considerable excursions. For composite n up to 5000, the minimum of 0.01471... occurs for n = 4489, and the maximum of 0.80849... occurs for n = 2310. This apparently large relative difference is actually surprisingly small: on log_2 scale it amounts to just about 5.78; a tiny fraction compared to the full scale, given by the values of n for the extrema. This insight suggests the following conjecture: there exists an average ratio r, defined as r = lim_{n->infinity} Sum_{m=1..n} r(m)/n. Its value appears to be approximately 0.3915+-0.0010, which can be interpreted as the average drop in a binomial value upon the 'reduction' of its arguments.