A391668 Table read by antidiagonals. T(n,k) is the least number coprime to all numbers in [n+1, n+k].

3, 5, 2, 5, 5, 3, 7, 7, 3, 2, 7, 7, 7, 7, 5, 11, 11, 11, 11, 5, 2, 11, 11, 11, 11, 5, 3, 3, 11, 11, 11, 11, 5, 5, 5, 2, 11, 11, 11, 11, 11, 11, 7, 7, 3, 13, 13, 13, 13, 13, 13, 7, 7, 3, 2, 13, 13, 13, 13, 13, 13, 7, 7, 7, 5, 5, 17, 17, 17, 17, 17, 17, 7, 7, 7, 5, 5, 2

Offset

1,1

References

Paul Erdős, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.

Paul Erdős and Ronald Graham, Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).

Links

Table of n, a(n) for n=1..78.

Thomas Bloom, Problem #663, Erdős Problems.

Thomas Bloom, Problem #457, Erdős Problems.

S. A. Burr, P. Erdős, R. L. Graham, and W. Wen-Ching Li, Complete sequences of sets of integer powers, Acta Arith. (1996), 133-138.

Formula

T(n,k) < (1 + o(1)) * k * log(n).

T(n,k) = A007918(n+k+1) for k >= n.

For all m < k, T(m*k,k) - T(m*k,k-1) > 0 iff k is prime.

Conjecture of Erdős and Pomerance: T(n,k) < (1+o(1)) * log(n), for fixed k >= 2 and sufficiently large n.

Conjecture of Erdős and Pomerance: there exists epsilon>0, such that T(n,log(n)) >= (2+epsilon)*log(n) infinitely often.

Example

The table begins:
  3,  5,  5,  7,  7, 11, 11, 11, 11, 13...
  2,  5,  7,  7, 11, 11, 11, 11, 13, 13...
  3,  3,  7, 11, 11, 11, 11, 13, 13, 17...
  2,  7, 11, 11, 11, 11, 13, 13, 17, 17...
  5,  5,  5,  5, 11, 13, 13, 17, 17, 17...
  2,  3,  5, 11, 13, 13, 17, 17, 17, 17...
  3,  5,  7,  7,  7,  7, 17, 17, 17, 19...
  2,  7,  7,  7,  7, 17, 17, 17, 19, 19...
  3,  3,  7,  7, 17, 17, 17, 19, 19, 23...
  2,  5,  5,  5, 17, 17, 19, 19, 23, 23...

Mathematica

pr=Prime@Range[1000];
q[n_, k_]:=Min[Complement[pr, Union@@Table[Divisors[n+i], {i, 1, k}]]];
Table[q[n, k], {n, 1, 200}, {k, 1, 200}]

Crossrefs

Cf. A053669, A053670, A053671, A053672, A053673 (columns).

Cf. A007918 (rows), A060264 (main diagonal).

Sequence in context: A090396 A086387 A073264 * A198099 A016657 A372143

Adjacent sequences: A391665 A391666 A391667 * A391669 A391670 A391671

Keyword

nonn,tabl

Author

Elijah Beregovsky, Dec 16 2025

Status

approved