A055441 - OEIS

%I #24 Apr 08 2026 10:27:45

%S 1,2,3,4,1,5,6,7,1,2,8,1,3,9,2,1,4,5,1,2,3,1,6,7,1,2,4,3,1,8,2,1,5,9,

%T 1,3,4,2,1,6,7,1,2,3,5,1,4,8,1,2,6,1,3,2,1,9,4,5,1,2,3,1,7,6,1,2,4,3,

%U 1,8,5,2,1,9,1,3,2,4,1,7,6,1,2,5,3,1,8,4,1,2,1,3,7,2,1,5,6,1,4,2,9,3,1,8,1

%N Distribution of first digit of mantissa following Benford's law, using largest remainder method with Hare quotas.

%C From _Pontus von Brömssen_, Mar 31 2026: (Start)

%C The apportionment method used here is an iterative version of the largest remainder method with Hare quotas, successively allocating 1 item (or seat) at a time to the element k maximizing n*p(k)-c(k), where p(k) is the probability mass of k and c(k) is the number of occurrences of k among the previous terms. This method is also used in A241773, for example. In some cases, the result differs from the non-iterative version (the Hamilton method). The latter method first allocates c(k) = floor(n*p(k)) items to element k; thereafter, the remaining items (if any) are allocated to elements k in decreasing order of the remainders n*p(k)-c(k). With this method, it may happen that the number of items allocated to a certain element decreases when n increases (the Alabama paradox); obviously, that does not happen for the iterative method, because no items are ever removed.

%C For the distribution considered here (Benford's law), the allocations of the iterative and non-iterative versions differ for n = 9, 26, 49, 52, 65, 82, 83, 89, 95, ... . It appears that all these cases are due to the Alabama paradox occurring in the Hamilton method, i.e., the number of items allocated to some element k, when n items are available, is smaller than it is for some smaller number of items. For the first 9 terms, for example, the numbers of occurrences of 1-9 in this sequence are (3, 1, 1, 1, 1, 1, 1, 0, 0), but the Hamilton method would give (3, 2, 1, 1, 1, 1, 0, 0, 0).

%C (End)

%H Pontus von Brömssen, <a href="/A055441/b055441.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Connelly, <a href="https://web.archive.org/web/20030803204148/http://www.solent.ac.uk/socsci/jc/voting/glossary.html">Glossary of voting terms</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BenfordsLaw.html">Benford's Law</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hare_quota">Hare quota</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quota_method">Quota method</a>.

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>.

%e a(50) = 2 so that after 50 terms we have had 15 1's, 9 2's, 6 3's, 5 4's, 4 5's, 3 6's, 3 7's, 3 8's and 2 9's.

%Y Cf. A055439, A055440, A055442, A241773.

%K nonn,base

%O 1,2

%A _Henry Bottomley_, May 17 2000