Matthias Engelhardt - cp's OEIS frontend

A181500 Triangle read by rows: number of solutions of n queens problem for given n and given number of queens engaged in conflicts.

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 28, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 8, 32, 48, 32

Offset: 0

Matthias Engelhardt, Oct 30 2010

Schlude and Specker investigate if it is possible to set n-1 non-attacking queens on an n X n toroidal chessboard. That is equivalent to searching for normal (i.e., non-toroidal) solutions of 3 engaged queens. In this case, one of the three queens has conflicts with both other queens. If you remove this queen, you get a setting of n-1 queens without conflicts, i.e., a toroidal solution.

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 0, 0, 2;
  10, 0, 0, 0, 0, 0;
   0, 0, 0, 0, 4, 0,  0;
  28, 0, 0, 0, 0, 0, 12, 0;
... - _Andrew Howroyd_, Dec 31 2017
For n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. For both solutions, all 4 queens are engaged in conflicts. So the terms for n=4 are 0 (0 solutions for n=4 having 0 engaged queens), 0, 0, 0 and 2 (the two cited above). These are members 11 to 15 of the sequence.

M. Engelhardt, Rows n=0..16 of triangle, flattened
Matthias Engelhardt, Conflicts in the n-queens problem
Matthias Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.
Konrad Schlude and Ernst Specker, Zum Problem der Damen auf dem Torus, Technical Report 412, Computer Science Department ETH Zurich, 2003.

Cf. A181499, A181501, A181502.

Row sum = A000170 (number of n-queen placements).
Column 0 has same values as A007705 (torus n-queen solutions).
Columns 1 and 2 are always zero.
Column 3 counts solutions of the special "Schlude-Specker" situation.

Offset corrected by Andrew Howroyd, Dec 31 2017

A181499 Triangle read by rows: number of solutions of n queens problem for given n and given number of conflicts.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 8, 4, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 0, 96, 24, 0, 0, 0, 0, 0, 0, 240, 0, 372, 112, 0, 0, 0, 0, 0, 88, 0, 0, 328, 1252, 872, 140, 0, 0, 0, 0, 0, 0, 0, 0, 3016, 5140, 4696, 1316, 32, 0, 0, 0, 0, 0

Offset: 0

Matthias Engelhardt, Oct 25 2010

			For n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

Matthias Engelhardt, Table of n, a(n) for n = 0..152 (corrected by Michel Marcus, Jan 19 2019)
M. Engelhardt, Conflicts in the n-queens problem
M. Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.

Cf. A000170, A007705, A181500, A181501, A181502.

Row sum = A000170 (number of n queens placements)
Column 0 has same values as A007705 (torus n queens solutions)
Column 1 is always zero.

A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 28, 0, 4, 8, 0, 0, 0, 0, 0, 0, 92, 0, 0, 0, 0, 0, 0, 0, 8, 272, 56, 16, 0, 0, 0, 0, 0, 0, 96, 344, 240, 44, 0, 0, 0, 0, 0, 0

Offset: 0

Matthias Engelhardt, Oct 30 2010

The rightmost part of the triangle contains only zeros. As any connection component needs at least two queens, the number of connection components of a solution is always less than or equal to n.

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 2, 0, 0;
  10, 0, 0, 0, 0, 0;
   0, 4, 0, 0, 0, 0, 0;
  28, 0, 4, 8, 0, 0, 0, 0;
  ... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two connection components in the conflicts graph. So, the terms for n=4 are 0, 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

M. Engelhardt, Rows n=0..16 of triangle, flattened
Matthias Engelhardt, Conflicts in the n-queens problem
Matthias Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.

Cf. A000170, A007705, A181499, A181500, A181502.

Row sum =A000170 (number of n queens placements)

Column 0 has same values as A007705 (torus n queens solutions)

Offset corrected by Andrew Howroyd, Dec 31 2017

A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 8, 4, 0, 0, 0, 0, 0, 0, 64, 24, 4, 0, 0, 0, 0, 0, 0, 248, 80, 16, 8, 0, 0, 0, 0, 0, 0, 172, 484, 36, 32, 0, 0, 0

Offset: 0

Matthias Engelhardt, Oct 30 2010

Torus solutions, i.e. solutions having an empty conflict constellation, are counted in column 1; this is caused by an interpretation of a queen not engaged in any conflict as an island in the conflict graph. Using the definition strictly, these queens should be removed from the graph and the numbers should appear in column 0, not column 1.

			Triangle begins:
  0;
  0,  1;
  0,  0, 0;
  0,  0, 0, 0;
  0,  0, 2, 0, 0;
  0, 10, 0, 0, 0, 0;
  0,  0, 0, 0, 4, 0, 0;
  0, 28, 8, 4, 0, 0, 0, 0;
... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

M. Engelhardt, Rows n=0..16 of triangle, flattened
Matthias Engelhardt, Conflicts in the n-queens problem
Matthias Engelhardt, Conflict tables for the n-queens problem
M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.

Cf. A000170, A007705, A181499, A181500, A181501.

Row sum =A000170 (number of n queens placements)
Column 1 has same values as A007705 (torus n queens solutions)
Column 0 is always zero.

Offset corrected by Andrew Howroyd, Dec 31 2017

A091350 First occurrence (*2) of n in A088627 - or - least number that yields n different primes if you factorize it in all possible ways in two factors and add these factors.

Original entry on oeis.org

8, 2, 6, 90, 30, 390, 690, 420, 210, 4290, 3990, 8778, 2310, 3570, 4830, 11550, 38850, 84630, 66990, 79170, 39270, 30030, 51870, 46410, 43890, 111930, 163020, 221340, 419430, 131670, 1902810, 1385670, 1009470, 1452990, 746130, 903210, 570570, 1067430, 1531530

Offset: 0

Matthias Engelhardt, Jan 05 2004

nonn

a(0) .. a(29) are in the list; additional know values are a(34) = 746130, a(35) = 903210, a(36) = 570570, a(40) = 510510, a(41) = 690690 and a(46) = 870870. If n in { 30, 31, 32, 33, 37, 38, 39, 42, 43, 44, 45}, or if n > 46, then a(n) > 10^6.

a(258) > 10^11. - Donovan Johnson, Oct 15 2013

			Sequence A088627 starts with 1,1,2,0, meaning that 2 and 4 yield 1 prime, 6 yields 2 and 8 yields 0 primes; therefore a(0) = 8, a(1) = 2 and a(2) = 6.

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..257 (terms up to a(90) from T. D. Noe)
M. Engelhardt, Number of Primes arising as Sum of a Factorization.

Cf. A088627.

// Programs available from Matthias Engelhardt.

DivPrimes[n_Integer] := Length[Select[Union[Divisors[n]+Reverse[Divisors[n]]], PrimeQ]]; nn=40; t=Table[0,{nn}]; cnt=0; k=0; While[cntT. D. Noe, Aug 02 2010 *)

Extended by T. D. Noe, Aug 02 2010

A062165 Number of ways of placing n nonattacking (normal) queens on n X n board, solutions similar on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 4, 13, 36, 115, 813, 3083, 21001, 131859, 868613

Offset: 1

Matthias Engelhardt

Two n-queens solutions p and q are considered similar iff there is a factor f, 0 < f < n, satisfying gcd (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n).

The sequence reduces exactly the objects of A062164 and, via that sequence, these of A002562 and A000170. Note that the equivalence classes of this sequence are a subset of A062168.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A062166 Number of ways of placing n nonattacking torus queens on 2n+1 X 2n+1 board, similar solutions count only once.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 7, 0, 26, 46, 0, 2861, 40303, 0, 6446047

Offset: 1

Matthias Engelhardt

Two n-queens solutions p and q are considered similar iff there is a factor f, 0 < f < n, satisfying gcd (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n).

The sequence reduces exactly the objects of A053994 and, via that sequence, these of A007705.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A062167 Number of permutations with at most 2 queens on any torus diagonal, solutions congruent on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 5, 29, 93, 569, 3226, 28630, 221250, 2314650

Offset: 1

Matthias Engelhardt

This sequence counts classes of "near n-queens solutions". Permutations with at most 1 queen on any torus diagonal are exactly the torus n queen solutions (A007705), those with at most 2 contain the normal n queen solutions (A000170).

Therefore they may be called "near n-queens solutions". In this sequence, permutations p and q are considered equivalent iff there are natural x and y, such that, for all k from {0, ..., n-1}, q (k + x mod n) = p (k) + y mod n, or q is a rotation or a reflection of such a q. In other words, rotations, reflections and torus shifts are allowed. The sequence contains the objects of A062164.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A062168 Number of permutations with at most 2 queens on any torus diagonal, solutions similar on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 3, 19, 31, 328, 650, 14542, 37449, 771570

Offset: 1

Matthias Engelhardt

As A062167, also this sequence counts classes of "near n-queens solutions". In this sequence, two permutations p and q are considered similar iff there is a factor f, 0 < f < n, satisfying GCD (f,n) = 1, such that for all k from {0, ..., n-1} q (k * f mod n) = p (k) * f mod n or if q is a rotation, a reflection or a shift of such a q. In other words, also expansions are allowed which move the queen at (k, p(k)) to (f * k mod n, f * p(k) mod n). The sequence contains the objects of A062165.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A062164 Number of ways of placing n nonattacking (normal) queens on n X n board; solutions congruent on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 6, 20, 40, 191, 953, 4604, 24660, 158466, 1009395

Offset: 1

Matthias Engelhardt

In this sequence two n-queens solutions p and q are considered equivalent iff there are natural numbers x and y such that, for all k from {0, ..., n-1}, q (k + x mod n) = p (k) + y mod n, or q is a rotation or a reflection of such a q.
In other words, besides rotations and reflections, also torus shifts are allowed. The sequence reduces the objects of A002562 and via that of A000170. The reduction of A000170 to this sequence is exactly the same as from A007705 to A053994 for torus queens; however, a solution for torus queens remains always a solution after a shift while a normal queens solutions does so only sometimes.
Note that the equivalence classes of this sequence are a subset of A006841. Moreover they are a subset of A062167.

M. Engelhardt, The N queens problem

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

cp's OEIS Frontend

User: Matthias Engelhardt

A181500 Triangle read by rows: number of solutions of n queens problem for given n and given number of queens engaged in conflicts.

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A181499 Triangle read by rows: number of solutions of n queens problem for given n and given number of conflicts.

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A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation.

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A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

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A091350 First occurrence (*2) of n in A088627 - or - least number that yields n different primes if you factorize it in all possible ways in two factors and add these factors.

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A062165 Number of ways of placing n nonattacking (normal) queens on n X n board, solutions similar on the torus count only once.

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A062166 Number of ways of placing n nonattacking torus queens on 2n+1 X 2n+1 board, similar solutions count only once.

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A062167 Number of permutations with at most 2 queens on any torus diagonal, solutions congruent on the torus count only once.

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A062168 Number of permutations with at most 2 queens on any torus diagonal, solutions similar on the torus count only once.

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