This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A011772 is applied to them, one obtains values [1, 3, 2, 7, 3, 8, 8, 8, 8], thus there are four divisors that obtain the maximal value 8 obtained at 36 itself, of which divisor 9 is the smallest, and therefore a(36) = 9.
A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772 A344758(n) = { my(x=A011772(n)); fordiv(n,d,if(A011772(d)==x, return(d))); }; (Python 3.8+) from itertools import combinations from math import prod from sympy import factorint, divisors from sympy.ntheory.modular import crt def A011772(n): plist = [p**q for p, q in factorint(2*n).items()] if len(plist) == 1: return n-1 if plist[0] % 2 else 2*n-1 return min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//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))) def A344758(n): m = A011772(n) for d in divisors(n): if A011772(d) == m: return d # Chai Wah Wu, Jun 03 2021
Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n], ds = Divisors[n]}, Do[If[s[d] == sn, Return[n/d]], {d, ds}]]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)
A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005 A354999(n) = { my(x=A344005(n)); fordiv(n, d, if(A344005(d)==x, return(n/d))); };