A080982 Smallest k such that the k-th triangular number has n^2 as divisor.

1, 7, 8, 31, 24, 8, 48, 127, 80, 24, 120, 63, 168, 48, 99, 511, 288, 80, 360, 224, 98, 120, 528, 512, 624, 168, 728, 735, 840, 224, 960, 2047, 242, 288, 49, 1215, 1368, 360, 675, 1024, 1680, 440, 1848, 1088, 324, 528, 2208, 512, 2400, 624, 288, 1183, 2808, 728

Offset: 1

Author

Reinhard Zumkeller, Feb 26 2003

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Crossrefs

Cf. A000217, A011772, A080983.

Programs

Haskell

import Data.List (findIndex)
import Data.Maybe (fromJust)
a080982 n = (+ 1) $ fromJust $
   findIndex ((== 0) . (`mod` (n ^ 2))) $ tail a000217_list
-- Reinhard Zumkeller, Mar 23 2013
(Python 3.8+)
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A080982(n):
    k = 2*n**2
    plist = [p**q for p, q in factorint(k).items()]
    return k-1 if len(plist) == 1 else min(min(crt([m,k//m],[0,-1])[0],crt([k//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l))) # Chai Wah Wu, Jun 13 2021

Formula

a(2^k) = 2^(2*k+1) - 1.

a(m) = m^2 - 1 for odd prime powers m.

A080983(n) = A000217(a(n)).