A139326 -id:A139326 - cp's OEIS frontend

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

James Propp, Jun 05 2003

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

			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

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.

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.

A084927 generalizes this to three dimensions.
Cf. A083415, A083383, A066888, A092556, A092557. See A083414 for primes in columns.
Cf. A139326.
Cf. A000720, A010051, A014085.

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

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;

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 *)

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

Edited by Charles R Greathouse IV, Jul 07 2010

A139325 Triangle read by rows: T(n,k) = number of primes in the k-th row of the square of the first n^2 odd numbers written consecutively in rows of length n, 1<=k<=n.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 4, 4, 2, 4, 2, 3, 5, 3, 4, 3, 3, 4, 2, 5, 5, 4, 3, 4, 2, 5, 2, 6, 4, 5, 4, 4, 4, 2, 4, 3, 7, 4, 5, 5, 3, 5, 4, 3, 4, 5, 7, 6, 4, 5, 6, 3, 4, 4, 5, 2, 6, 8, 6, 5, 4, 6, 4, 5, 4, 4, 5, 4, 5, 8, 6, 6, 6, 4, 5, 6, 4, 5, 4, 6, 3, 4, 8, 7, 7, 6, 5, 5, 5, 4, 6, 5, 4, 4, 5, 5

Offset: 1

Reinhard Zumkeller, Apr 14 2008

A139326(n) = Min{T(n,k): 1<=k<=n};
A139327(n) = Max{T(n,k): 1<=k<=n};
A139328(n) gives sums of rows;
T(n,1) = A099802(n) - 1.

Cf. A083415.

A139327 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = maximal number of primes in a row.

Original entry on oeis.org

0, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 17, 17, 17, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 29, 29, 29, 29, 29, 30, 30, 31, 31, 31, 32, 33, 33, 33, 33, 33

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Max{A139325(n,k): 1<=k<=n}.

			a(4)=Max{#{3,5,7},#{11,13},#{17,19,23},#{29,31}}=Max{3,2,3,2}=3:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A139326.

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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.

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A139325 Triangle read by rows: T(n,k) = number of primes in the k-th row of the square of the first n^2 odd numbers written consecutively in rows of length n, 1<=k<=n.

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A139327 Write the first n^2 odd numbers consecutively in n rows of length n: a(n) = maximal number of primes in a row.

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