A083486 Triangle read by rows in which the n-th row contains the smallest set of n distinct numbers beginning with n with a product which is a square.

1, 2, 8, 3, 4, 12, 4, 5, 6, 30, 5, 6, 7, 8, 105, 6, 7, 8, 9, 10, 210, 7, 8, 9, 10, 11, 12, 1155, 8, 9, 10, 11, 12, 13, 14, 30030, 9, 10, 11, 12, 13, 14, 15, 16, 1001, 10, 11, 12, 13, 14, 15, 16, 17, 18, 34034, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 323323, 12, 13, 14, 15, 16, 17

Offset

1,2

Comments

Presumably "smallest set" means we start with n-1 consecutive numbers in each row and add the last element to satisfy the requirement on the square, the last element is obtained from the prime factorization of the product of the first n-1 numbers by reducing all prime exponents modulo 2. If the result is <= the (n-1)-st term, multiply by j^2, j=2,3,4,.. until it is. - R. J. Mathar, Apr 05 2007

Links

Table of n, a(n) for n=1..72.

Example

1
2 8
3 4 12
4 5 6 30
5 6 7 8 105
...

Maple

A083486 := proc(n, m) local fs, k, resu, extr ; if m < n then n+m-1; else fs := ifactors(mul( A083486(n, k), k=1..n-1))[2] ; resu := mul( op(1, op(k, fs))^(op(2, op(k, fs)) mod 2), k=1..nops(fs)) ; extr := 1 ; while extr^2*resu <= A083486(n, n-1) do extr := extr+1 ; od ; RETURN(resu*extr^2) ; fi ; end: for n from 1 to 15 do for m from 1 to n do printf("%a ", A083486(n, m)) ; od ; od ; # R. J. Mathar, Apr 05 2007

Mathematica

T[n_, m_] := Module[{fs, resu, extr }, If[m < n, n+m-1, fs = FactorInteger[Product[T[n, k], {k, 1, n-1}]]; resu = Product[fs[[k, 1]]^Mod[fs[[k, 2]], 2], {k, 1, Length[fs]}]; extr = 1; While[extr^2*resu <= T[n, n-1], extr++]; resu*extr^2]];
Table[T[n, m], {n, 1, 15}, {m, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

Crossrefs

Cf. A083484, A083485.

Sequence in context: A016640 A379829 A321984 * A076031 A076596 A081967

Adjacent sequences: A083483 A083484 A083485 * A083487 A083488 A083489

Keyword

nonn,tabl

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

Extensions

Corrected and extended by R. J. Mathar, Apr 05 2007

Example corrected by Harvey P. Dale, Aug 05 2021

Status

approved