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

0,2

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

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)

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]

Prog

(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

Crossrefs

Cf. A000217, A033996, A175497, A306415.

Sequence in context: A122812 A022763 A244370 * A033996 A333714 A146980

Adjacent sequences: A179679 A179680 A179681 * A179683 A179684 A179685

Keyword

nonn,look

Author

Robert G. Wilson v, Jul 24 2010

Extensions

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

Status

approved