A055441 Distribution of first digit of mantissa following Benford's law, using largest remainder method with Hare quotas.

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, 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, 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

Offset

1,2

Comments

From Pontus von Brömssen, Mar 31 2026: (Start)

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.

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).

(End)

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

J. Connelly, Glossary of voting terms.

Eric Weisstein's World of Mathematics, Benford's Law.

Wikipedia, Hare quota.

Wikipedia, Quota method.

Index entries for sequences related to Benford's law.

Example

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.

Crossrefs

Cf. A055439, A055440, A055442, A241773.

Sequence in context: A253146 A253028 A133108 * A104717 A067003 A277884

Adjacent sequences: A055438 A055439 A055440 * A055442 A055443 A055444

Keyword

nonn,base

Author

Henry Bottomley, May 17 2000

Status

approved