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

1,9

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

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.

Example

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

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 *)

Prog

(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
(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

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.

Sequence in context: A082478 A279060 A324119 * A327168 A329037 A279794

Adjacent sequences: A083379 A083380 A083381 * A083383 A083384 A083385

Keyword

nonn,nice

Author

James Propp, Jun 05 2003

Extensions

Edited by Charles R Greathouse IV, Jul 07 2010

Status

approved