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A354048 a(n) is the largest number of distinct integer-sided right triangles in which some n-digit number can appear as the length of a side.

%I A354048 #36 Mar 22 2025 19:02:46
%S A354048 2,14,68,203,476,1421,3293,7910,20060,39509,89324,206711,442907,
%T A354048 803924,1722464,3198608,6820523,13434254,27901259,50222267
%N A354048 a(n) is the largest number of distinct integer-sided right triangles in which some n-digit number can appear as the length of a side.
%H A354048 Zhihu, <a href="https://www.zhihu.com/question/531237744">For integers from 1 to 100, which one can compose the most Pythagorean triangle?</a>
%e A354048 a(2)=14 because there exist 14 distinct integer-sided right triangles with the 2-digit number 60 as the length of a side, i.e., (11,60,61), (25,60,65), (32,60,68), (36,48,60), (45,60,75), (60,63,87), (60,80,100), (60,91,109), (60,144,156), (60,175,185), (60,221,229), (60,297,303), (60,448,452), and (60,899,901), and no 2-digit number is the length of a side of more than 14 distinct integer-sided right triangles.
%o A354048 (Python)
%o A354048 from sympy import factorint
%o A354048 def s(n):
%o A354048     f=factorint(n)
%o A354048     d, q=(list(f.keys()), list(f.values()))
%o A354048     (a, b, c, x)=(0, 1, 1, 0)
%o A354048     if(d[0]==2):
%o A354048         a, x=(0, 1)
%o A354048         if q[0]>1:
%o A354048              a=q[0]-1
%o A354048     for p in range(x, len(d)):
%o A354048         b*=(1+2*q[p])
%o A354048         if d[p]%4==1:
%o A354048             c*=(1+2*q[p])
%o A354048     return((b-1)//2+a*b+(c-1)//2)
%o A354048 def a(n):
%o A354048     max=0
%o A354048     for i in range(1+10**(n-1), 10**n):
%o A354048         if s(i)>max:
%o A354048             k,max=(i,s(i))
%o A354048     return(n,[k,max])
%o A354048 for i in range(1,6):
%o A354048     print (a(i))
%o A354048 # (thanks to _Zhao Hui Du_ for help in the derivation of this function)
%Y A354048 Cf. A046081, A269929, A353875.
%K A354048 nonn,more
%O A354048 1,1
%A A354048 _Zhining Yang_, Jun 26 2022