A081454 Triangle read by rows in which the n-th row contains n distinct numbers whose product is a square, which is minimal over all choices for n distinct numbers.

1, 1, 4, 1, 2, 8, 1, 2, 3, 6, 1, 2, 3, 4, 6, 1, 2, 3, 4, 6, 9, 1, 2, 3, 5, 6, 8, 10, 1, 2, 3, 4, 5, 6, 8, 10, 1, 2, 3, 4, 5, 6, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 22

Offset

1,3

Comments

In case there is more than one solution, choose the one where the maximal number is minimal.

Links

Max Alekseyev, Rows n = 1..50, flattened

Example

Triangle begins:
  1;
  1,  4;
  1,  2,  8;
  1,  2,  3,  6;
  1,  2,  3,  4,  6;
  1,  2,  3,  4,  6,  9;
  1,  2,  3,  5,  6,  8, 10;
  1,  2,  3,  4,  5,  6,  8, 10;
  1,  2,  3,  4,  5,  6,  8,  9, 10;
  ...
The 7th row could also be 1, 2, 3, 4, 5, 8, 15, but this has a larger last term.

Maple

A081454aux := proc(n, s, mfact) local d, findx, f ; if n = 1 then if s <= mfact then RETURN([s]) ; else RETURN([]) ; fi ; else d := numtheory[divisors](s) ; for findx from n to nops(d) do if op(findx, d) <= mfact then f := A081454aux(n-1, s/op(findx, d), op(findx, d)-1) ; if nops(f) <> 0 then RETURN([op(f), op(findx, d)]) ; fi ; fi ; od ; RETURN([]) ; fi ; end: A081454row := proc(n) local p, s, d, findx, f ; p :=1 ; s :=1 ; while true do d := numtheory[divisors](s) ; if nops(d) >= n then if n = 1 then RETURN([1]) ; else for findx from n to nops(d) do f := A081454aux(n-1, s/op(findx, d), op(findx, d)-1) ; if nops(f) <> 0 then RETURN([op(f), op(findx, d)]) ; fi ; od; fi ; fi ; p := p+1 ; s := p^2 ; od ; end: for n from 1 to 14 do r := A081454row(n) : for i from 1 to n do printf("%d, ", op(i, r) ) ; od ; od : # R. J. Mathar, Nov 12 2006

Mathematica

T[n_] := T[n] = SortBy[MinimalBy[Select[Subsets[Range[2n+2], {n}], #[[1]] == 1 && IntegerQ@Sqrt[Times @@ #]&], Times @@ #&], Last] // First;
Table[Print[n, " ", T[n]]; T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jun 03 2023 *)

Crossrefs

Cf. A081455, A081456, A081457.

Sequence in context: A216863 A120578 A096249 * A133883 A279816 A004160

Adjacent sequences: A081451 A081452 A081453 * A081455 A081456 A081457

Keyword

tabl,nonn

Author

Amarnath Murthy, Mar 21 2003

Extensions

Edited and extended by David Garber, Jun 17 2003

More terms from Ray G. Opao, Aug 01 2005

Corrected and extended by R. J. Mathar, Nov 12 2006

More terms from Max Alekseyev, Apr 25 2009

Correct row #13 conjectured by Jean-François Alcover and confirmed by Max Alekseyev, Jun 03 2023

Status

approved