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

1,1

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.

Links

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

Example

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.

Prog

(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

Crossrefs

Cf. A006278, A236381, A061060.

Sequence in context: A171459 A240558 A163896 * A262698 A347946 A168054

Adjacent sequences: A350810 A350811 A350812 * A350814 A350815 A350816

Keyword

nonn

Author

Richard Peterson, Jan 17 2022

Extensions

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

Status

approved