A232891 Least positive integer m <= n^2/2 + 3 such that {k*prime(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, 7, 7, 10, 17, 43, 13, 51, 22, 51, 36, 31, 49, 64, 71, 119, 73, 86, 68, 141, 110, 153, 85, 83, 86, 144, 81, 174, 127, 115, 87, 122, 138, 143, 134, 133, 142, 211, 229, 152, 104, 109, 177, 259, 142, 194, 176, 196, 311, 312, 193, 243, 197, 396, 169, 156, 171

Offset

1,2

Comments

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

(ii) For any positive integer n not equal to 3, the number n*prime(n) + 1 cannot be a power x^m with m > 1.

(iii) There are infinitely many positive integers n with n - 1, n + 1, n + prime(n), n + prime(n)^2, n^2 + prime(n), n^2 + prime(n)^2 all prime.

Links

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

Example

a(3) = 4 since 1*prime(1) = 2, 2*prime(2) === 3*prime(3) == 0 (mod 3), and 4*prime(4) = 28 == 1 (mod 3).

Mathematica

L[m_, n_]:=Length[Union[Table[Mod[k*Prime[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, " ", counterexample]; Label[aa]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000040, A232616, A232861, A232862.

Sequence in context: A360794 A220848 A370605 * A096223 A232862 A344460

Adjacent sequences: A232888 A232889 A232890 * A232892 A232893 A232894

Keyword

nonn

Author

Zhi-Wei Sun, Dec 02 2013

Status

approved