A390003 Irregular triangle read by rows: row n lists all the quadruples (a,b,c,d) such that a!*b!*c!*d! is a perfect square, with 1 <= a < b < c < d = A387184(n). Within a row, quadruples are sorted by increasing a, then by increasing b, and finally by increasing c.

1, 3, 5, 6, 1, 4, 5, 6, 1, 2, 7, 8, 1, 2, 7, 9, 3, 4, 8, 9, 1, 6, 7, 10, 2, 6, 8, 10, 2, 6, 9, 10, 3, 5, 7, 10, 4, 5, 7, 10, 2, 3, 11, 12, 2, 4, 11, 12, 2, 10, 13, 14, 6, 8, 13, 14, 6, 9, 13, 14, 2, 5, 14, 15, 3, 7, 13, 15, 4, 7, 13, 15, 5, 10, 13, 15

Offset

1,2

Links

Table of n, a(n) for n=1..76.

Formula

T(n,4) = A387184(n).

Example

Triangle begins:
  (1,  3,  5,  6), (1,  4,  5,  6);
  (1,  2,  7,  8);
  (1,  2,  7,  9), (3,  4,  8,  9);
  (1,  6,  7, 10), (2,  6,  8, 10), (2,  6,  9, 10), (3,  5,  7, 10), (4,  5,  7, 10);
  (2,  3, 11, 12), (2,  4, 11, 12);
  (2, 10, 13, 14), (6,  8, 13, 14), (6,  9, 13, 14);
  (2,  5, 14, 15), (3,  7, 13, 15), (4,  7, 13, 15), (5, 10, 13, 15);
  (2,  5, 14, 16), (3,  4, 15, 16), (3,  7, 13, 16), (4,  7, 13, 16), (5, 10, 13, 16), ( 8,  9, 15, 16);
  (1,  2, 17, 18), (7,  8, 17, 18), (7,  9, 17, 18);
  (1,  6, 19, 20), (3,  5, 19, 20), (4,  5, 19, 20), (7, 10, 19, 20);
  (3,  8, 19, 21), (3,  9, 19, 21), (4,  8, 19, 21), (4,  9, 19, 21), (5,  8, 20, 21), ( 5,  9, 20, 21);
  (2, 12, 20, 22), (3, 11, 20, 22), (4, 11, 20, 22), (5, 11, 19, 22);
  (1,  3, 23, 24), (1,  4, 23, 24), (5,  6, 23, 24);
  (1,  3, 23, 25), (1,  4, 23, 25), (3,  4, 24, 25), (5,  6, 23, 25), (8,  9, 24, 25), (15, 16, 24, 25);
  ...

Mathematica

Table[Module[{row = {}}, If[CompositeQ[d], Do[If[IntegerQ[Sqrt[d!/c!*b!/a!]], AppendTo[row, {a, b, c, d}]], {c, NextPrime[d, -1], d-1}, {b, 2, c-1}, {a, b-1}]]; If[row != {}, Sort[row], Nothing]], {d, 20}]

Crossrefs

Cf. A387184 (d values), A390004 (number of quadruples in each row), A390005.

Cf. A388851, A389553.

Sequence in context: A388508 A243589 A395437 * A178255 A154467 A152713

Adjacent sequences: A390000 A390001 A390002 * A390004 A390005 A390006

Keyword

nonn,tabf

Author

Paolo Xausa, Oct 21 2025

Status

approved