

A232178


Least k>=0 such that triangular(n) + k^2 is a square, or 1 if no such k exists.


4



0, 0, 1, 1, 1, 1, 2, 6, 0, 2, 3, 1, 1, 3, 4, 1, 15, 4, 5, 1, 1, 5, 6, 20, 10, 6, 7, 1, 1, 7, 8, 27, 1, 8, 9, 1, 1, 9, 10, 2, 36, 10, 11, 1, 1, 11, 12, 41, 7, 0, 13, 1, 1, 13, 6, 24, 2, 14, 15, 1, 1, 15, 16, 3, 6, 8, 17, 1, 1, 17, 18, 62, 64, 18, 19
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OFFSET

0,7


COMMENTS

Triangular(n) = n*(n+1)/2.


LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000


EXAMPLE

a(7) = 6 because the least k such that triangular(n) + k^2 is a square is k=6: 7*(7+1)/2 + 6^2 = 28+36 = 64 = 8^2.


MATHEMATICA

Join[{0}, Table[k = 0; While[k < n && ! IntegerQ[Sqrt[n*(n + 1)/2 + k^2]], k++]; If[k == n, k = 1]; k, {n, 100}]] (* T. D. Noe, Nov 21 2013 *)


PROG

(Python)
from __future__ import division
from sympy import divisors
def A232178(n):
if n == 0:
return 0
t = n*(n+1)//2
ds = divisors(t)
l, m = divmod(len(ds), 2)
if m:
return 0
for i in range(l1, 1, 1):
x = ds[i]
y = t//x
a, b = divmod(yx, 2)
if not b:
return a
return 1 # Chai Wah Wu, Sep 12 2017


CROSSREFS

Cf. A000217, A000290.
Cf. A082183 (least k>0 such that triangular(n) + triangular(k) is a triangular number).
Cf. A232177 (least k>0 such that triangular(n) + triangular(k) is a square).
Cf. A232176 (least k>0 such that n^2 + triangular(k) is a square).
Cf. A232179 (least k>=0 such that n^2 + triangular(k) is a triangular number).
Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).
Sequence in context: A072340 A118354 A080730 * A016590 A079461 A220233
Adjacent sequences: A232175 A232176 A232177 * A232179 A232180 A232181


KEYWORD

sign


AUTHOR

Alex Ratushnyak, Nov 20 2013


STATUS

approved



