

A267077


Least m>0 for which m*n^2 + 1 is a square and m*triangular(n) + 1 is a triangular number (A000217). Or 1 if no such m exists.


2



1, 35, 30, 18135, 189, 27, 321300, 23760, 1188585957, 1656083, 26, 244894427400, 82093908624206325, 1858717755529547, 86478, 21491811639746039592
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OFFSET

0,2


COMMENTS

a(18) = 26780050, a(20) = 274554988002, a(22) = 13519299793860, a(27) = 3713566235, a(28) = 329517, a(40) = 17185833329121, a(44) = 802273222814658, a(54) = 56877965914. For 16 <= n <= 100 that are not listed here, if a(n) > 0, then a(n) > 10^14.  Chai Wah Wu, Jan 18 2016


LINKS

Table of n, a(n) for n=0..15.


EXAMPLE

26*10^2+1 = 2601 is a square, and 26*10*11/2+1 = 1431 = triangular(53), and 26 is the smallest such multiplier, therefore a(10)=26.


PROG

(Python)
from math import sqrt
def A267077(n):
if n == 0:
return 1
u, v, t, w = max(8, 2*n), max(4, n)**29, 4*n*(n+1), n**2
while True:
m, r = divmod(v, t)
if not r and int(sqrt(m*w+1))**2 == m*w+1:
return m
v += u+1
u += 2 # Chai Wah Wu, Jan 15 2016


CROSSREFS

Cf. A000217, A000290, A035096, A061782, A067872, A188621.
Sequence in context: A174027 A259083 A244214 * A267394 A022991 A023477
Adjacent sequences: A267074 A267075 A267076 * A267078 A267079 A267080


KEYWORD

nonn,hard,more


AUTHOR

Alex Ratushnyak, Jan 10 2016


EXTENSIONS

a(12)a(15) from Chai Wah Wu, Jan 16 2016


STATUS

approved



