

A341578


a(n) is the minimum number of total votes needed for one party to win if there are n^2 voters divided into equal districts.


3



1, 3, 4, 8, 9, 14, 16, 24, 25, 33, 36, 45, 49, 60, 64, 80, 81, 95, 100, 117, 121, 138, 144, 165, 169, 189, 196, 224, 225, 247, 256, 288, 289, 315, 324, 350, 361, 390, 400, 429, 441, 473, 484, 528, 529, 564, 576, 624, 625, 663, 676, 728, 729, 770, 784, 825, 841, 885, 900, 943
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OFFSET

1,2


COMMENTS

Comments from Jack W Grahl and Andrew Weimholt, Feb 26 2021: (Start):
This is a twoparty election. The size d of each district must divide n^2, so there are n^2/d equal districts.
The districts are winnertakesall, and tied districts go to neither candidate. For an even number of districts, it is enough to win half the districts and tie in one further district.
So for 5 districts of 5 votes each, one party could win with 3 votes in each of 3 districts, and 0 in all other districts, for a total of a(5) = 9 votes.
For 8 districts of size 8, 5 votes in each of 4 districts and 4 votes in a fifth district are enough, for a total of a(8) = 24 votes.
d need not equal n. For n=6, it is better to gerrymander the 36 votes into 3 districts with 12 votes each, and then a(6) = 14 = 7+7+0 votes are enough to win. (End)
This is related to the gerrymandering question. What is the asymptotic behavior of a(n)?  N. J. A. Sloane, Feb 20 2021. Answer from Don Reble, Feb 26 2020: The lower bound is [(n^2+1)/4 + n/2]; the upper bound is [n^2/4 + n]. Each bound is reached infinitely often. In general the best choice for d is not unique, since d and n/d give the same answer.


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..5000
N. J. A. Sloane, How to use Gerrymandering to win with only 14 votes when you opponent has 22: (1) Locations of voters
N. J. A. Sloane, How to use Gerrymandering to win with only 14 votes when you opponent has 22: (2) Make three districts with 12 voters in each, drawn so that your 14 votes win 2 out of the 3 districts
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.


FORMULA

a(n) is the minimum value of {(floor(d/2)+1)*(floor(n^2/(2*d))+1) over all divisors d of n^2 AND (n/2+1)^21, if n is even}.


EXAMPLE

For a(2), divisors of 2^2 are 1, 2, 4:
d=1: (floor(1/2)+1)*(floor(2^2/(2*1))+1) = 1*3 = 3
d=3: (floor(2/2)+1)*(floor(2^2/(2*2))+1) = 2*2 = 4
d=9: (floor(4/2)+1)*(floor(2^2/(2*4))+1) = 3*1 = 3
OR
since n is even, ((2/2)+1)^21=3
Party A only needs 3 cells out of 4 to win a majority of districts.
For a(6), divisors of 6^2 are 1, 2, 3, 4, 6, 9, 12, 18, 36:
By symmetry we can ignore d = 9, 12, 18 and 36;
d=1: (floor(1/2)+1)*(floor(6^2/(2*1))+1) = 1*19 = 19
d=2: (floor(2/2)+1)*(floor(6^2/(2*2))+1) = 2*10 = 20
d=3: (floor(3/2)+1)*(floor(6^2/(2*3))+1) = 2*7 = 14
d=4: (floor(4/2)+1)*(floor(6^2/(2*4))+1) = 3*5 = 15
d=6: (floor(6/2)+1)*(floor(6^2/(2*6))+1) = 4*4 = 16
OR
since n is even, ((6/2)+1)^21=15
Party A only needs 14 cells out of 36 to win a majority of districts.


MATHEMATICA

Table[Min[Table[(Floor[d/2]+1)*(Floor[n^2/(2*d)]+1), {d, Divisors[n^2]}], If[EvenQ[n], (n/2+1)^21, Infinity]], {n, 60}](* Stefano Spezia, Feb 15 2021 *)


PROG

(Python)
from sympy import divisors
def A341578(n):
c = min((d//2+1)*(n**2//(2*d)+1) for d in divisors(n**2, generator=True) if d<=n)
return c if n % 2 else min(c, (n//2+1)**21) # Chai Wah Wu, Mar 05 2021


CROSSREFS

See A341721 for an analog where there are n voters, not n^2.
See A341319 for a variant.
See also A290323.
Sequence in context: A047204 A050035 A306901 * A046974 A177986 A186775
Adjacent sequences: A341575 A341576 A341577 * A341579 A341580 A341581


KEYWORD

nonn


AUTHOR

Sean Chorney, Feb 14 2021


EXTENSIONS

Entry revised by N. J. A. Sloane, Feb 26 2021.


STATUS

approved



