A083382 Write the numbers from 1 to n^2 consecutively in n rows of length n; a(n) = minimal number of primes in a row.

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 4, 3, 3, 4, 3, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 6, 4, 5, 6, 6, 5, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 7, 7, 7, 7, 6, 6, 7, 7, 7

Offset: 1

Author

James Propp, Jun 05 2003

Keywords

• nonn
• nice

Comments

Conjectured by Schinzel (Hypothesis H2) to be always positive for n > 1.

The conjecture has been verified for n = prime < 790000 by Aguilar.

If this is true, then Legendre's conjecture is true as well. (See A014085). - Antti Karttunen, Jan 01 2019

Examples

			For n = 3 the array is
1 2 3 (2 primes)
4 5 6 (1 prime)
7 8 9 (1 prime)
so a(3) = 1
		

References

• P. Ribenboim, The New Book of Prime Number Records, Chapter 6.
• P. Ribenboim, The Little Book Of Big Primes, Springer-Verlag, NY 1991, page 185.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Carlos Rivera, The calendar-like square conjecture
• A. Schinzel and W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.

Crossrefs

A084927 generalizes this to three dimensions.

Cf. A083415, A083383, A066888, A092556, A092557. See A083414 for primes in columns.

Cf. A139326.

Cf. A000720, A010051, A014085.

Programs

Haskell

a083382 n = f n n a010051_list where
   f m 0 _     = m
   f m k chips = f (min m $ sum chin) (k - 1) chips' where
     (chin,chips') = splitAt n chips
-- Reinhard Zumkeller, Jun 10 2012

Maple

A083382 := proc(n) local t1,t2,at; t1 := n; at := 0; for i from 1 to n do t2 := 0; for j from 1 to n do at := at+1; if isprime(at) then t2 := t2+1; fi; od; if t2 < t1 then t1 := t2; fi; od; t1; end;

Mathematica

Table[minP=n; Do[s=0; Do[If[PrimeQ[c+(r-1)*n], s++ ], {c, n}]; minP=Min[s, minP], {r, n}]; minP, {n, 100}]
Table[Min[Count[#,?PrimeQ]&/@Partition[Range[n^2],n]],{n,110}] (* _Harvey P. Dale, May 29 2013 *)

PARI

A083382(n) = { my(m=-1); for(i=0,n-1,my(s=sum(j=(i*n),((i+1)*n)-1,isprime(1+j))); if((m<0) || (s < m), m = s)); (m); }; \\ Antti Karttunen, Jan 01 2019

Extensions

Edited by Charles R Greathouse IV, Jul 07 2010