A083415 -id:A083415 - 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

A083414 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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 10 2003

nonn

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, Jun 10 2003
The conjecture is true for all n <= 40000.

			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.

See A083382 for references and links.

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A083415 and A083382 for primes in rows.
A084927 generalizes this to three dimensions.
Cf. A010051.

a083414 n = minimum $ map c $ filter ((== 1) . (gcd n)) [1..n] where
   c k = sum $ map a010051 $ enumFromThenTo k (k + n) (n ^ 2)
-- Reinhard Zumkeller, Jun 10 2012

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}]

More terms from Vladeta Jovovic and T. D. Noe, Jun 10 2003

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.

A092556 Triangle read by rows: T(1,1) = 1; for n>=2, write the first n^2 integers in an n X n array beginning with 1 in the upper left proceeding left to right and top to bottom; then T(n,k) is the first prime in the k-th row.

Original entry on oeis.org

1, 2, 3, 2, 5, 7, 2, 5, 11, 13, 2, 7, 11, 17, 23, 2, 7, 13, 19, 29, 31, 2, 11, 17, 23, 29, 37, 43, 2, 11, 17, 29, 37, 41, 53, 59, 2, 11, 19, 29, 37, 47, 59, 67, 73, 2, 11, 23, 31, 41, 53, 61, 71, 83, 97, 2, 13, 23, 37, 47, 59, 67, 79, 89, 101, 113, 2, 13, 29, 37, 53, 61, 73, 89, 97, 109

Offset: 1

Robert G. Wilson v, Feb 27 2004

There is a prime in each row.

Paulo Ribenboim, "The Little Book Of Big Primes," Springer-Verlag, NY 1991, page 185.

Cf. A092557, A083415.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim[i*n], {n, 2, 12}, {i, 0, n - 1}]

A092557 Triangle read by rows: T(1,1) = 1; for n>=2, write the first n^2 integers in an n X n array beginning with 1 in the upper left proceeding left to right and top to bottom; then T(n,k) is the last prime in the k-th row.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 3, 7, 11, 13, 5, 7, 13, 19, 23, 5, 11, 17, 23, 29, 31, 7, 13, 19, 23, 31, 41, 47, 7, 13, 23, 31, 37, 47, 53, 61, 7, 17, 23, 31, 43, 53, 61, 71, 79, 7, 19, 29, 37, 47, 59, 67, 79, 89, 97, 11, 19, 31, 43, 53, 61, 73, 83, 97, 109, 113, 11, 23, 31, 47, 59, 71, 83, 89

Offset: 1

Robert G. Wilson v, Feb 27 2004

There is a prime in each row.

			Triangle begins
1;
2, 3;
3, 5, 7;
3, 7, 11, 13;
5, 7, 13, 19, 23;

Paulo Ribenboim, "The Little Book Of Big Primes," Springer-Verlag, NY 1991, page 185.

Cf. A092556, A083415.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; Table[ PrevPrim[i*n + 1], {n, 2, 12}, {i, 1, n}]

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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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A083414 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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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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A092556 Triangle read by rows: T(1,1) = 1; for n>=2, write the first n^2 integers in an n X n array beginning with 1 in the upper left proceeding left to right and top to bottom; then T(n,k) is the first prime in the k-th row.

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A092557 Triangle read by rows: T(1,1) = 1; for n>=2, write the first n^2 integers in an n X n array beginning with 1 in the upper left proceeding left to right and top to bottom; then T(n,k) is the last prime in the k-th row.

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