A084927 If the numbers 1 to n^3 are arranged in a cubic array, a(n) is the minimum number of primes in each column of the n^2 columns in the "top view" that can have primes.

0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 0, 3, 1, 1, 0, 4, 0, 3, 1, 3, 0, 8, 0, 2, 2, 3, 1, 5, 0, 2, 1, 4, 0, 9, 0, 3, 2, 4, 0, 6, 1, 6, 2, 4, 0, 5, 0, 5, 2, 3, 0, 11, 0, 4, 3, 3, 1, 10, 1, 5, 3, 7, 0, 10, 0, 2, 4, 6, 2, 11, 1, 7, 3, 5, 0, 13, 2, 6, 4, 7, 1, 17, 2, 6, 2, 6, 2, 12, 1, 8, 4, 8

Offset: 1

Author

T. D. Noe, Jun 12 2003

Keywords

• nonn

Comments

This is a three-dimensional generalization of A083414.

Examples

			For the case n=3, the numbers are arranged in a cubic array as follows:
1..2..3........10.11.12........19.20.21
4..5..6........13.14.15........22.23.24
7..8..9........16.17.18........25.26.27
The first column is (1,10,19), the second is (2,11,20), etc. Only columns whose tops are relatively prime to n are counted. In this case, columns starting with 3, 6 and 9 cannot have primes. a(n) = 0 for n = 1, 25, 55 and the primes from 5 to 83, except 67 and 79. It appears that a(n) > 0 for n > 83. This has been confirmed up to n = 1000.
		

References

• See A083382 for references and links to the two-dimensional case.

Crossrefs

Cf. A083382, A083414, A084928 (east-west view), A084929 (north-south view).

Programs

Mathematica

Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r-1)*n^2], s++ ], {r, n}]; minP=Min[s, minP]], {c, n^2}]; minP, {n, 100}]