A081977 -id:A081977 - cp's OEIS frontend

A081974 a(1) = 1 and smallest number not occurring earlier such that the product of two neighboring terms is a distinct triangular number, where "distinct" means that a(n)*a(n+1) may not equal the product of any two previous consecutive terms.

1, 3, 2, 5, 9, 4, 7, 13, 6, 11, 21, 10, 12, 23, 45, 14, 27, 53, 26, 30, 33, 16, 31, 15, 20, 41, 81, 40, 52, 103, 51, 25, 49, 24, 47, 93, 46, 91, 60, 76, 151, 75, 37, 18, 112, 115, 57, 28, 55, 39, 19, 87, 43, 22, 133, 66, 58, 117, 35, 17, 8, 395, 126, 141, 70, 102, 90, 116, 231

Offset: 1

Amarnath Murthy, Apr 03 2003

nonn

Perhaps another re-arrangement of natural numbers.

			5 and 4 are the neighbors of 9 giving the triangular numbers 45 and 36 respectively.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A081975, A081976, A081977.

Acands:= [$2..10^4]: ncands:= 10^4-1: Tused:= {}:
a[1]:= 1:
for i from 2 to 100 do
  found:= false;
  for j from 1 to ncands do
    if istri(a[i-1]*Acands[j]) and not member(a[i-1]*Acands[j],Tused) then
       found:= true; a[i]:= Acands[j];
       Acands:= subsop(j=NULL, Acands);
       ncands:= ncands - 1;
       Tused:= Tused union {a[i-1]*a[i]};
       break
    fi
   od;
   if not found then break fi
od:
seq(a[k],k=1..100); # Robert Israel, Jul 23 2025

istriang[n_] := With[{x = Floor[Sqrt[2*n]]}, n == x*(x + 1)/2];
nmax = 75;
Clear[a, used, tris];
a[] = 0; used[] = 0; tris[_] = 0; a[1] = 1; used[1] = 1;
For[i = 2, i <= nmax, i++, f = a[i-1]; j = 2; While[used[j] == 1 || !istriang[f*j] || tris[f*j] == 1, j++]; a[i] = j; used[j] = 1; tris[f*j] = 1];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, May 23 2024, after PARI code *)

istriang(n) = local(x); x = floor(sqrt(2*n)); n == x*(x + 1)/2;
A = vector(75); used = vector(1000); tris = vector(50000); A[1] = 1; used[1] = 1; for (i = 2, 75, f = A[i - 1]; j = 2; while (used[j] || !istriang(f*j) || tris[f*j], j = j + 1); A[i] = j; used[j] = 1; tris[f*j] = 1); print(A)

More terms from David Wasserman, Jul 26 2004

A081975 Triangular number pertaining to A081974. a(n) = A081974(n)*A081974(n+1).

Original entry on oeis.org

3, 6, 10, 45, 36, 28, 91, 78, 66, 231, 210, 120, 276, 1035, 630, 378, 1431, 1378, 780, 990, 528, 496, 465, 300, 820, 3321, 3240, 2080, 5356, 5253, 1275, 1225, 1176, 1128, 4371, 4278, 4186, 5460, 4560, 11476, 11325, 2775, 666, 2016, 12880, 6555, 1596

Offset: 1

Amarnath Murthy, Apr 03 2003

nonn

A subset of A000217. - R. J. Mathar, Apr 05 2007

Cf. A081974, A081976, A081977.

isA000217 := proc(n) local t; t := (sqrt(1+8*n)-1)/2 ; type(t,'integer'); end: A081974 := proc(nmax) local a,n,prodset; a := [1,3] ; prodset := {3} ; while nops(a) < nmax do n := 2 ; while n in a or n*op(-1,a) in prodset or isA000217(n*op(-1,a)) = false do n := n+1 ; od ; prodset := prodset union { n*op(-1,a) } ; a := [op(a),n] ; od ; RETURN(a) ; end: A081975 := proc(nmax) local a ; a081974 := A081974(nmax) ; a := [] ; for i from 2 to nops(a081974) do a := [op(a), op(i,a081974)*op(i-1,a081974)] ; od ; RETURN(a) ; end: a := A081975(100) ; # R. J. Mathar, Apr 05 2007

istriang[n_] := With[{x = Floor[Sqrt[2*n]]}, n == x*(x + 1)/2];
nmax = 47;
Clear[b, used, tris];
b[] = 0; used[] = 0; tris[_] = 0; b[1] = 1; used[1] = 1;
For[i = 2, i <= nmax+1, i++, f = b[i-1]; j = 2; While[used[j] == 1 || !istriang[f*j] || tris[f*j] == 1, j++]; b[i] = j; used[j] = 1; tris[f*j] = 1];
a[n_] := b[n]*b[n + 1];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, May 23 2024, after PARI code in A081974 *)

More terms from R. J. Mathar, Apr 05 2007

cp's OEIS Frontend

A081974 a(1) = 1 and smallest number not occurring earlier such that the product of two neighboring terms is a distinct triangular number, where "distinct" means that a(n)*a(n+1) may not equal the product of any two previous consecutive terms.

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