A386313 Excess for a circular sparse ruler, A283297(n) - (A000194(n-1) + 1).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1

Comments

Terms of A283297[n] (fewest marks for a circular sparse ruler) closely follow A000194[n] (round(sqrt(n))).

Term a(133)=0, for Singer "perfect partition" {1, 2, 9, 8, 14, 4, 43, 7, 6, 10, 5, 24}. The current best known value for a(132)=2.

Links

Table of n, a(n) for n=1..131.

MathOverflow, What is the smallest cardinality of a self-linked set in a finite cyclic group?, Feb 15 2017.

MathStackExchange, Sparse Cyclic Sum sets, Jul 13 2025.

Formula

For prime powers q, a(q^2 + q + 1) = 0.

Example

For length 117, either sparse ruler {0, 1, 4, 8, 10, 22, 27, 36, 47, 60, 76, 91} or cyclic sum set {1, 3, 4, 2, 12, 5, 9, 11, 13, 16, 15, 26} has 12 marks or 12 parts. The excess is 12 - (round(sqrt(117 - 1)) + 1) = 0
  {{101, 12, {0,1,10,18,22,39,41,44,50,57,77,87}},
   {102, 12, {0,1,4,16,19,40,42,50,59,64,70,77}},
   {103, 12, {0,1,3,17,29,31,50,56,61,65,83,96}},
   {104, 12, {0,1,3,15,17,42,46,55,65,72,83,99}},
   {105, 12, {0,1,3,12,20,34,38,53,60,66,76,81}},
   {106, 12, {0,1,3,9,19,39,53,61,66,83,90,94}},
   {107, 12, {0,1,9,19,24,31,52,56,58,69,72,98}},
   {108, 12, {0,1,3,15,21,46,50,59,69,76,87,103}},
   {109, 12, {0,1,10,12,14,17,32,40,61,67,85,86}},
   {110, 12, {0,1,5,7,15,24,33,36,57,68,70,95}},
   {111, 12, {0,1,3,15,35,46,60,64,73,83,90,106}},
   {112, 12, {0,1,7,9,21,35,40,53,57,82,98,109}},
   {113, 12, {0,1,7,15,20,24,27,53,55,71,92,103}}}.

Crossrefs

Cf. A000194, A283297, A326499.

Sequence in context: A324731 A011734 A276653 * A355443 A394742 A011733

Adjacent sequences: A386310 A386311 A386312 * A386314 A386315 A386316

Keyword

nonn,hard

Author

Ed Pegg Jr, Jul 18 2025

Status

approved