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Write the numbers from 1 to n^2 consecutively in n rows of length n; a(n) = minimal number of primes in a row.
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%I #24 Jan 02 2019 11:54:12

%S 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,

%T 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,

%U 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

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

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

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

%C If this is true, then Legendre's conjecture is true as well. (See A014085). - _Antti Karttunen_, Jan 01 2019

%D P. Ribenboim, The New Book of Prime Number Records, Chapter 6.

%D P. Ribenboim, The Little Book Of Big Primes, Springer-Verlag, NY 1991, page 185.

%H Charles R Greathouse IV, <a href="/A083382/b083382.txt">Table of n, a(n) for n = 1..10000</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_026.htm">The calendar-like square conjecture</a>

%H A. Schinzel and W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf">Sur certaines hypothèses concernant les nombres premiers</a>, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.

%e For n = 3 the array is

%e 1 2 3 (2 primes)

%e 4 5 6 (1 prime)

%e 7 8 9 (1 prime)

%e so a(3) = 1

%p 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;

%t 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}]

%t Table[Min[Count[#,_?PrimeQ]&/@Partition[Range[n^2],n]],{n,110}] (* _Harvey P. Dale_, May 29 2013 *)

%o (Haskell)

%o a083382 n = f n n a010051_list where

%o f m 0 _ = m

%o f m k chips = f (min m $ sum chin) (k - 1) chips' where

%o (chin,chips') = splitAt n chips

%o -- _Reinhard Zumkeller_, Jun 10 2012

%o (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

%Y A084927 generalizes this to three dimensions.

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

%Y Cf. A139326.

%Y Cf. A000720, A010051, A014085.

%K nonn,nice

%O 1,9

%A _James Propp_, Jun 05 2003

%E Edited by _Charles R Greathouse IV_, Jul 07 2010