A380377 -id:A380377 - cp's OEIS frontend

A380380 Numbers k such that A380377(k)/k (proportion of supporters needed to win an election when there are k voters) sets a new minimum.

1, 3, 4, 7, 10, 13, 16, 17, 23, 24, 31, 38, 45, 49, 59, 60, 67, 71, 82, 93, 97, 104, 111, 112, 123, 127, 142, 157, 161, 172, 180, 195, 199, 218, 229, 237, 241, 256, 264, 283, 287, 310, 325, 333, 337, 356, 364, 379, 387, 391, 418, 437, 445, 449, 472, 480, 499

Offset: 1

Pontus von Brömssen, Jan 24 2025

nonn

The only term common to this sequence and A380379 is 1.

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

Cf. A380376, A380377, A380379.

A380379 Least k for which A380377(k) = n, i.e., the least number of voters for which n supporters are needed to win an election with rules as in A380377.

Original entry on oeis.org

1, 2, 5, 8, 11, 14, 18, 20, 25, 28, 32, 33, 39, 40, 46, 50, 53, 54, 61, 62, 68, 72, 74, 75, 83, 86, 88, 94, 98, 99, 105, 106, 113, 116, 117, 124, 128, 129, 130, 138, 143, 144, 150, 151, 158, 162, 163, 164, 173, 176, 181, 182, 188, 189, 196, 200, 203, 204, 205

Offset: 1

Pontus von Brömssen, Jan 24 2025

nonn

Also, indices of records in A380377.

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

Cf. A380377, A380380.

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

Pontus von Brömssen, Jan 24 2025

nonn

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

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Illustration for a(100000)=363.

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

Pontus von Brömssen, Jan 24 2025

nonn

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

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Illustration for a(100000)=551.

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

Pontus von Brömssen, Jan 24 2025

342 is almost certainly the last term of this sequence.

Numbers k such that A380377(k) = A380378(k,m) for some even m.

			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.

Cf. A380377, A380378.

A380378 Triangle read by rows: T(n,k) is the minimum number of total votes needed for one party to win if there are n voters divided into k balanced districts, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 3, 4, 4, 5, 5, 4, 5, 4, 4, 4, 5, 5, 5, 4, 5, 5, 4, 4, 5, 5, 6, 6, 4, 5, 6, 5, 4, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 5, 5, 6, 6, 7, 7, 6, 6, 6, 7, 6, 5, 5, 6, 6, 7, 7, 7, 6, 6, 6, 7, 7, 6, 5, 6, 6, 7, 7

Offset: 1

Pontus von Brömssen, Jan 24 2025

See A380377 for further details.

It is never optimal to have any supporters in a losing district or to win a district with a greater margin than necessary. This implies that, in any optimal strategy, any district of size m should have 0, m/2, (m+1)/2, or m/2+1 supporters. If k is odd, the optimal strategy is to win the (k+1)/2 smallest districts. If k is even and n/k is an odd integer, the best strategy is to win k/2+1 districts (all districts have n/k voters in this case). If k is even and n/k is not an odd integer, the best strategy is to draw one of the even districts and win the k/2 smallest remaining districts.

			Triangle begins:
  n\k| 1  2  3  4  5  6  7  8  9 10 11 12
  ---+-----------------------------------
   1 | 1
   2 | 2  2
   3 | 2  2  2
   4 | 3  3  2  3
   5 | 3  3  3  3  3
   6 | 4  4  4  3  3  4
   7 | 4  4  4  4  3  4  4
   8 | 5  5  4  5  4  4  4  5
   9 | 5  5  4  5  5  4  4  5  5
  10 | 6  6  4  5  6  5  4  5  5  6
  11 | 6  6  5  5  6  6  5  5  5  6  6
  12 | 7  7  6  6  6  7  6  5  5  6  6  7

Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 rows)
Pontus von Brömssen, Illustration for row n=100000.

Cf. A380377 (row minima), A380379, A380380, A380381, A380382, A380383.

def A380378(n,k):
    q,r = divmod(n,k)
    q2,rq = divmod(q,2)
    k2,rk = divmod(k,2)
    x = (k2+1)*(q2+1)
    if 2*r

cp's OEIS Frontend

A380380 Numbers k such that A380377(k)/k (proportion of supporters needed to win an election when there are k voters) sets a new minimum.

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A380379 Least k for which A380377(k) = n, i.e., the least number of voters for which n supporters are needed to win an election with rules as in A380377.

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

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

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

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A380378 Triangle read by rows: T(n,k) is the minimum number of total votes needed for one party to win if there are n voters divided into k balanced districts, 1 <= k <= n.

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