cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A380377 Minimum number of total votes needed for one party to win if there are n voters divided into balanced districts, i.e., the numbers of voters in two districts may differ by at most 1.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22
Offset: 1

Views

Author

Pontus von Brömssen, Jan 24 2025

Keywords

Comments

The rules are the same as in A341721 (except that the number of voters in two districts may differ by 1 here): The winner must have a strict majority of the votes in a strictly larger number of districts than the other party has.
Empirically, it seems that the limit of (a(n)-n/4)/sqrt(n) exists with an approximate value of 0.3538.

Examples

			For n = 9, a(9) = 4 votes are required to win. There can be either 3 districts 3+3+3 with 2 supporters in 2 of them, 6 districts 1+1+1+2+2+2 with 3 supporters in the single-voter districts and 1 in a 2-voter district, or 7 districts 1+1+1+1+1+2+2 with supporters in 4 of the single-voter districts.
For n = 17, a(17) = 6 votes are required to win. This can only be achieved with 5 districts 3+3+3+4+4 with 2 supporters in each of the 3 smaller districts.

Links

Crossrefs

Row minima of A380378.
Cf. A341721, A380379, A380380, A380381, A380382, A380383.

Formula

a(n) <= A341721(n).
a(n) = a(n-1)+1 if n is in A380379, otherwise a(n) = a(n-1).
a(n) = A380378(n,A380381(n)) = A380378(n,A380382(n)).

A380381 Smallest number of districts needed for A380377(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 5, 3, 3, 3, 3, 8, 9, 3, 3, 3, 5, 5, 13, 3, 3, 3, 7, 7, 5, 5, 5, 3, 9, 9, 9, 9, 3, 3, 5, 5, 5, 7, 7, 3, 12, 12, 13, 13, 13, 5, 5, 9, 9, 7, 7, 7, 7, 5, 5, 5, 5, 11, 11, 11, 11, 7, 7, 7, 7, 7, 9, 9, 13, 13, 13, 13, 21, 21, 5, 5, 5, 7, 7, 7, 11
Offset: 1

Views

Author

Pontus von Brömssen, Jan 24 2025

Keywords

Comments

Smallest k such that A380378(n,k) = A380377(n).

Links

Crossrefs

Cf. A380377, A380378, A380382 (largest number of districts).

A380382 Largest number of districts possible for A380377(n).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 11, 11, 11, 5, 13, 13, 15, 15, 15, 7, 7, 17, 5, 5, 19, 9, 9, 9, 9, 23, 23, 11, 11, 11, 11, 11, 27, 13, 13, 13, 13, 13, 13, 9, 9, 9, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 11, 11, 19, 19, 19, 19, 19, 9, 13, 13, 13, 13, 21
Offset: 1

Views

Author

Pontus von Brömssen, Jan 24 2025

Keywords

Comments

Largest k such that A380378(n,k) = A380377(n).

Links

Crossrefs

Cf. A380377, A380378, A380381 (smallest number of districts).

A380383 Numbers k such that the minimum number of votes A380377(k) can be attained with an even number of districts.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 18, 20, 21, 28, 33, 34, 35, 40, 41, 42, 54, 55, 56, 62, 63, 75, 76, 77, 88, 99, 130, 131, 132, 164, 165, 206, 207, 208, 209, 210, 300, 341, 342
Offset: 1

Views

Author

Pontus von Brömssen, Jan 24 2025

Keywords

Comments

342 is almost certainly the last term of this sequence.
Numbers k such that A380377(k) = A380378(k,m) for some even m.

Examples

			A380377(342) = 95 can be obtained with either 36 districts 18*9 + 18*10 with 5 supporters in each of the 18 9-voter districts and 5 supporters in one of the 10-voter districts, or 37 districts 28*9 + 9*10 with 5 supporters in 19 of the 9-voter districts. Since one solution has an even number of districts, 342 is a term.

Crossrefs

Cf. A380377, A380378.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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