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A083414
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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.
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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 (noe(AT)sspectra.com), Jun 10 2003
The conjecture is true for all n <= 40000.
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REFERENCES
| See A083382 for references and links.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..2000
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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.
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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}]
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CROSSREFS
| Cf. A083415 and A083382 for primes in rows.
A084927 generalizes this to three dimensions.
Sequence in context: A072064 A105498 A179289 * A171174 A171173 A163618
Adjacent sequences: A083411 A083412 A083413 * A083415 A083416 A083417
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 10 2003
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and T. D. Noe (noe(AT)sspectra.com), Jun 10 2003
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