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A145299
Smallest k such that k^2+1 is divisible by A002144(n)^6.
8
1068, 1999509, 390112, 253879357, 756360062, 2363588163, 5041394261, 9435321777, 41865466758, 102666405913, 197177418061, 316411915250, 171829799914, 625667121807, 182312430890, 1095001339019, 6390289199260
OFFSET
1,1
EXAMPLE
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.
PROG
(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
CROSSREFS
Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145296, A145297, A145298.
Sequence in context: A234880 A218565 A145298 * A328894 A289702 A095673
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Oct 17 2008
EXTENSIONS
More terms and efficient PARI program from. - Max Alekseyev, Oct 28 2008
STATUS
approved