%I #22 Sep 29 2025 23:24:04

%S 1,7,8,31,24,8,48,127,80,24,120,63,168,48,99,511,288,80,360,224,98,

%T 120,528,512,624,168,728,735,840,224,960,2047,242,288,49,1215,1368,

%U 360,675,1024,1680,440,1848,1088,324,528,2208,512,2400,624,288,1183,2808,728

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

%H Chai Wah Wu, <a href="/A080982/b080982.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Reinhard Zumkeller)

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

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

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

%o (Haskell)

%o import Data.List (findIndex)

%o import Data.Maybe (fromJust)

%o a080982 n = (+ 1) $ fromJust $

%o findIndex ((== 0) . (`mod` (n ^ 2))) $ tail a000217_list

%o -- _Reinhard Zumkeller_, Mar 23 2013

%o (Python)

%o from itertools import combinations

%o from sympy import factorint

%o from sympy.ntheory.modular import crt

%o def A080982(n):

%o k = 2*n**2

%o plist = [p**q for p, q in factorint(k).items()]

%o 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

%Y Cf. A000217, A011772, A080983.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Feb 26 2003