A179682 Least integer, k, greater than n such that t(k)*t(n) form a perfect square; t(i) is the i-th triangular number (A000217).

1, 8, 24, 48, 80, 120, 168, 224, 49, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 242, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280, 8648

Offset: 0

Author

Robert G. Wilson v, Jul 24 2010

Keywords

• nonn
• look

Comments

It appears that a(n) = A033996(n) for most n. - Robert Israel, Feb 15 2019

Links

• Robert Israel, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000217, A033996, A175497, A306415.

Programs

Maple

f:= proc(n) local F,t,p,k0,d,k,a,j;
  p:= max(map(t -> `if`(t[2]::odd, t[1],NULL), [op(ifactors(n)[2]),op(ifactors(n+1)[2])]));
  if n mod p = 0 then k0:= n+p-1; d:= 1;
    else  k0:= n+1; d:= p-1;
  fi;
  t:= n*(n+1)/4;
  for a from k0 by p do
    for k in [a, a+d] do
       if issqr(k*(k+1)*t) then return k fi
  od od
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Feb 15 2019

Mathematica

f[n_] := Block[{k = n + 1, n2 = n (n + 1)/2}, While[ !IntegerQ@ Sqrt[n2*k (k + 1)/2], k++ ]; k]; Array[f, 47, 0]

Python

from sympy.ntheory.primetest import is_square
def A179682(n):
    m = n*(n+1)>>1
    k = n+1
    while not is_square(m*k*(k+1)>>1):
        k += 1
    return k # Chai Wah Wu, Mar 13 2023

Formula

From Robert Israel, Feb 15 2019: (Start)

a(n) <= A033996(n).

If n = A033996(j) then a(n) <= A033996(a(j)).

If n = a(j) < A033996(j) then a(n) <= A033996(j).

(End)

Extensions

Incorrect empirical g.f. removed by Robert Israel, Feb 15 2019