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.

2, 14, 68, 203, 476, 1421, 3293, 7910, 20060, 39509, 89324, 206711, 442907, 803924, 1722464, 3198608, 6820523, 13434254, 27901259, 50222267

Offset: 1

Author

Zhining Yang, Jun 26 2022

Keywords

• nonn
• more

Examples

			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.
		

Links

• Zhihu, For integers from 1 to 100, which one can compose the most Pythagorean triangle?

Crossrefs

Cf. A046081, A269929, A353875.

Programs

Python

from sympy import factorint
def s(n):
    f=factorint(n)
    d, q=(list(f.keys()), list(f.values()))
    (a, b, c, x)=(0, 1, 1, 0)
    if(d[0]==2):
        a, x=(0, 1)
        if q[0]>1:
             a=q[0]-1
    for p in range(x, len(d)):
        b*=(1+2*q[p])
        if d[p]%4==1:
            c*=(1+2*q[p])
    return((b-1)//2+a*b+(c-1)//2)
def a(n):
    max=0
    for i in range(1+10**(n-1), 10**n):
        if s(i)>max:
            k,max=(i,s(i))
    return(n,[k,max])
for i in range(1,6):
    print (a(i))
# (thanks to Zhao Hui Du for help in the derivation of this function)