A145299 Smallest k such that k^2+1 is divisible by A002144(n)^6.

1068, 1999509, 390112, 253879357, 756360062, 2363588163, 5041394261, 9435321777, 41865466758, 102666405913, 197177418061, 316411915250, 171829799914, 625667121807, 182312430890, 1095001339019, 6390289199260

Offset: 1

Author

Klaus Brockhaus, Oct 17 2008

Keywords

• nonn

Examples

			a(1) = 1068 since A002144(1) = 5, 1068^2+1 = 1140625 = 5^6*73 and for no k < 1068 does 5^6 divide k^2+1. a(11) = 197177418061 since A002144(11) = 97, 197177418061^2+1 = 38878934193202368999722 = 2*97^6*23337479509 and for no k < 197177418061 does 97^6 divide k^2+1.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298.

Programs

PARI

{ e=6; forprime(p=2, 1000, if(p%4==1, k=lift(sqrt(-1+O(p^e))); if(k>p^e/2,k=p^e-k); print1(k, ", "))) }

Python

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145299_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**6))
A145299_list = list(islice(A145299_gen(),20)) # Chai Wah Wu, May 04 2024

Extensions

More terms and efficient PARI program from. - Max Alekseyev, Oct 28 2008