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.

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, 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, 15, 12, 14, 9, 26, 6, 15, 12, 23, 5, 25, 3, 15, 13, 17, 8, 29, 7, 20, 12, 17, 7, 32

Offset: 1

Author

N. J. A. Sloane, Jun 10 2003

Keywords

• nonn

Comments

Conjectured to be always positive for n>1.

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

The conjecture is true for all n <= 40000.

Examples

			For n = 4 the array is
.   1  2  3  4
.   5  6  7  8
.   9 10 11 12
.  13 14 15 16
in which columns 1 and 3 contain 2 and 3 primes; therefore a(4) = 2.
		

References

• See A083382 for references and links.

Links

• T. D. Noe, Table of n, a(n) for n=1..2000

Crossrefs

Cf. A083415 and A083382 for primes in rows.

A084927 generalizes this to three dimensions.

Cf. A010051.

Programs

Haskell

a083414 n = minimum $ map c $ filter ((== 1) . (gcd n)) [1..n] where
   c k = sum $ map a010051 $ enumFromThenTo k (k + n) (n ^ 2)
-- Reinhard Zumkeller, Jun 10 2012

Mathematica

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}]

Extensions

More terms from Vladeta Jovovic and T. D. Noe, Jun 10 2003