A232862 Least positive integer m <= n^2/2 + 3 such that the set {prime(k) - k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

1, 3, 4, 11, 9, 8, 10, 15, 29, 13, 23, 22, 23, 37, 32, 28, 48, 44, 53, 41, 45, 67, 76, 117, 119, 91, 121, 88, 89, 101, 72, 88, 100, 143, 144, 185, 145, 104, 176, 141, 144, 175, 187, 213, 121, 255, 128, 129, 189, 243, 122, 267, 275, 242, 209, 205, 130, 153, 263, 335

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0.

Note that a(4) = 11 = 4^2/2 + 3.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Zhi-Wei Sun)

Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Example

a(3) = 4 since prime(1) - 1 = prime(2) - 2 = 1, prime(3) - 3 = 2, prime(4) - 4 = 3, and {1,2,3} is a complete system of residues modulo 3.

Mathematica

  L[m_, n_]:=Length[Union[Table[Mod[Prime[k]-k, n], {k, 1, m}]]]
Do[Do[If[L[m, n]==n, Print[n, " ", m]; Goto[aa]], {m, 1, n^2/2+3}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 60}]

Crossrefs

Cf. A014689, A232616, A232463, A232548, A232861.

Sequence in context: A370605 A232891 A096223 * A344460 A195589 A339578

Adjacent sequences: A232859 A232860 A232861 * A232863 A232864 A232865

Keyword

nonn

Author

Zhi-Wei Sun, Dec 01 2013

Status

approved