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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).
2
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
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
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