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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 (list; graph; refs; listen; history; text; internal format)
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)**2-9, 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

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Last modified November 29 11:36 EST 2020. Contains 338765 sequences. (Running on oeis4.)