

A232176


Least positive k such that n^2 + triangular(k) is a square.


4



1, 2, 6, 10, 14, 18, 7, 5, 8, 34, 6, 42, 46, 15, 54, 16, 14, 66, 70, 74, 23, 82, 9, 90, 17, 98, 102, 10, 110, 15, 25, 122, 126, 16, 39, 48, 40, 21, 150, 34, 158, 29, 54, 48, 30, 13, 182, 63, 55, 194, 56, 202, 14, 45, 214, 63, 222, 26, 41, 234, 31, 42, 39, 250, 32, 63
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Triangular(k) = A000217(k) = k*(k+1)/2.
a(n) <= 4*n  2, because with k = 4*n2: n^2 + k*(k+1)/2 = n^2 + (4*n2)*(4*n1)/2 = 9*n^2  6*n + 1 = (3*n1)^2.
The sequence of numbers n such that a(n)=n begins: 8, 800, 7683200 ...  a subsequence of A220186.


LINKS

Table of n, a(n) for n=0..65.


MATHEMATICA

lpk[n_]:=Module[{k=1}, While[!IntegerQ[Sqrt[n^2+(k(k+1))/2]], k++]; k]; Array[ lpk, 70, 0] (* Harvey P. Dale, May 04 2018 *)


PROG

(Python)
import math
for n in range(77):
n2 = n*n
y=1
for k in range(1, 10000001):
sum = n2 + k*(k+1)/2
r = int(math.sqrt(sum))
if r*r == sum:
print str(k)+', ',
y=0
break
if y: print ', ',
(PARI) a(n) = {k = 1; while (! issquare(n^2 + k*(k+1)/2), k++); k; } \\ Michel Marcus, Nov 20 2013


CROSSREFS

Cf. A000290, A000217, A038202, A055527, A220186, A232175.
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).
Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).
Sequence in context: A067368 A191259 A184914 * A187884 A068977 A251538
Adjacent sequences: A232173 A232174 A232175 * A232177 A232178 A232179


KEYWORD

nonn


AUTHOR

Alex Ratushnyak, Nov 19 2013


STATUS

approved



