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%I #4 Mar 30 2012 17:22:28
%S 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,
%T 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,
%U 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
%N 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.
%C This is a three-dimensional generalization of A083414.
%D See A083382 for references and links to the two-dimensional case.
%e For the case n=3, the numbers are arranged in a cubic array as follows:
%e 1..2..3........10.11.12........19.20.21
%e 4..5..6........13.14.15........22.23.24
%e 7..8..9........16.17.18........25.26.27
%e 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.
%t 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}]
%Y Cf. A083382, A083414, A084928 (east-west view), A084929 (north-south view).
%K nonn
%O 1,6
%A _T. D. Noe_, Jun 12 2003
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