A350813 a(n) is the least positive number k such that the product of the first n primes that are congruent to 1 (mod 4) is equal to y^2 - k^2 for some integer y.

2, 4, 24, 38, 16, 588, 5782, 5528, 80872, 319296, 3217476, 32301914, 20085008, 166518276, 2049477188, 17443412442, 27905362944, 233647747282, 886295348972, 134684992249108, 98002282636962, 392994156083892, 5283713761100536, 76642755213473624, 923250078609721236

Offset: 1

Author

Richard Peterson, Jan 17 2022

Keywords

• nonn

Comments

Because y^2-k^2=(y-k)(y+k), a method to make k as small as possible is to try to make y-k and y+k as nearly equal as possible.

Because each of y-k and y+k are made up of primes of form 1 mod 4, algebra shows that k=a(n) is always even.

Examples

			For n=3, m = 5*13*17. The "middle" most nearly equal divisor and codivisor of m are y-k=17 and y+k=65, whence a(n) = (65 - 17)/2 = 24.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..36

Crossrefs

Cf. A006278, A236381, A061060.

Programs

Python

from math import prod, isqrt
from itertools import islice
from sympy import sieve, divisors
def A350813(n):
    m = prod(islice(filter(lambda p: p % 4 == 1, sieve),n))
    a = isqrt(m)
    d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
    return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022

Extensions

Terms corrected by and more terms from Jinyuan Wang, Mar 17 2022