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A083414 Write the numbers from 1 to n^2 consecutively in n rows of length n; let c(k) = number of primes in k-th column; a(n) = minimal c(k) for gcd(k,n) = 1.

%I A083414 #15 Jun 14 2013 04:07:27
%S A083414 0,1,1,2,1,4,1,2,3,5,2,6,1,5,5,5,2,10,2,6,5,8,3,9,5,8,5,9,4,17,3,9,7,
%T A083414 9,6,15,4,9,8,13,4,21,3,11,10,11,4,17,5,15,9,14,5,20,8,14,9,14,6,27,6,
%U A083414 15,12,14,9,26,6,15,12,23,5,25,3,15,13,17,8,29,7,20,12,17,7,32
%N A083414 Write the numbers from 1 to n^2 consecutively in n rows of length n; let c(k) = number of primes in k-th column; a(n) = minimal c(k) for gcd(k,n) = 1.
%C A083414 Conjectured to be always positive for n>1.
%C A083414 Note that a(n) is large when phi(n), the number of integers relatively prime to n, is small and vice versa. - _T. D. Noe_, Jun 10 2003
%C A083414 The conjecture is true for all n <= 40000.
%D A083414 See A083382 for references and links.
%H A083414 T. D. Noe, <a href="/A083414/b083414.txt">Table of n, a(n) for n=1..2000</a>
%e A083414 For n = 4 the array is
%e A083414 .   1  2  3  4
%e A083414 .   5  6  7  8
%e A083414 .   9 10 11 12
%e A083414 .  13 14 15 16
%e A083414 in which columns 1 and 3 contain 2 and 3 primes; therefore a(4) = 2.
%t A083414 Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r-1)*n], s++ ], {r, n}]; minP=Min[s, minP]], {c, n}]; minP, {n, 100}]
%o A083414 (Haskell)
%o A083414 a083414 n = minimum $ map c $ filter ((== 1) . (gcd n)) [1..n] where
%o A083414    c k = sum $ map a010051 $ enumFromThenTo k (k + n) (n ^ 2)
%o A083414 -- _Reinhard Zumkeller_, Jun 10 2012
%Y A083414 Cf. A083415 and A083382 for primes in rows.
%Y A083414 A084927 generalizes this to three dimensions.
%Y A083414 Cf. A010051.
%K A083414 nonn
%O A083414 1,4
%A A083414 _N. J. A. Sloane_, Jun 10 2003
%E A083414 More terms from _Vladeta Jovovic_ and _T. D. Noe_, Jun 10 2003