OFFSET
1,2
COMMENTS
The D'Hondt method is applicable even though the sum of the weights is infinite.
a(n) is the number k minimizing (c(k)+1)/p(k), where c(k) is the number of occurrences of k among the previous terms.
No ties occur. Proof: Assume that there is a tie between values j and k at some point. Then log(1+1/j)/log(1+1/k) = (c(j)+1)/(c(k)+1) = r/s is rational, which implies that (j+1)^s/j^s = (k+1)^r/k^r. Both these fractions are in lowest terms (since GCD(j,j+1) = GCD(k,k+1) = 1), so j^s = k^r and (j+1)^s = (k+1)^r, i.e., s*log(j) = r*log(k) and s*log(j+1) = r*log(k+1). This implies that log(j)/log(j+1) = log(k)/log(k+1). Since the function log(x)/log(x+1) is strictly increasing for x > 0, it follows that j = k.
A055440 is the subsequence of terms less than 10. In general, the subsequence of terms less than b is the sequence generated by the D'Hondt method applied to Benford's law for base b.
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Plot of A395154(n)*(log n)/n coloured by occurrence.
Wikipedia, Benford's law.
Wikipedia, D'Hondt method.
FORMULA
Positions of k are given by n_k(m) = Sum_{j>=1} floor(m*log(1+1/j)/log(1+1/k)), m >= 1. (Terms in the series are nonzero only for j <= 1/((1+1/k)^(1/m)-1).)
a(n)*(log n)/n goes to 1/m as n goes to infinity through values such that a(n) occurs for the m-th time, i.e., k*(log n_k(m))/n_k(m) goes to 1/m as k goes to infinity.
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Apr 14 2026
STATUS
approved