

A083414


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


6



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
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OFFSET

1,4


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.


REFERENCES

See A083382 for references and links.


LINKS

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


EXAMPLE

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.


MATHEMATICA

Table[minP=n; Do[If[GCD[c, n]==1, s=0; Do[If[PrimeQ[c+(r1)*n], s++ ], {r, n}]; minP=Min[s, minP]], {c, n}]; minP, {n, 100}]


PROG

(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


CROSSREFS

Cf. A083415 and A083382 for primes in rows.
A084927 generalizes this to three dimensions.
Cf. A010051.
Sequence in context: A072064 A105498 A179289 * A171174 A171173 A268671
Adjacent sequences: A083411 A083412 A083413 * A083415 A083416 A083417


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jun 10 2003


EXTENSIONS

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


STATUS

approved



