A145296 Smallest k such that k^2 + 1 is divisible by A002144(n)^3.

57, 239, 1985, 10133, 9466, 11389, 27590, 51412, 153765, 344464, 107551, 296344, 172078, 432436, 931837, 753090, 676541, 2321221, 2027724, 3394758, 1706203, 4841182, 1438398, 2947125, 398366, 5657795, 4942017, 9400802, 11906503

Offset: 1

Author

Klaus Brockhaus, Oct 08 2008

Keywords

• nonn

Examples

			a(3) = 1985 since A002144(3) = 17, 1985^2 + 1 = 3940226 = 2*17^3*401 and for no k < 1985 does 17^3 divide k^2+1.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..150 from Klaus Brockhaus)

Crossrefs

Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145297, A145298, A145299.

Programs

PARI

{m=12000000; pmax=300; z=70; v=vector(z); for(n=1, m, fac=factor(n^2+1); for(j=1, #fac[, 1], if(fac[j, 2]>=3&&fac[j, 1]<=pmax, q=primepi(fac[j, 1]); if(q<=z&&v[q]==0, v[q]=n)))); t=1; j=0; while(t&&j
				

PARI

{e=3; forprime(p=2, 300, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))} \\ Klaus Brockhaus, Oct 09 2008

Python

from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145296_gen(): # generator of terms
    p = 1
    while (p:=nextprime(p)):
        if p&3==1:
            yield min(sqrt_mod_iter(-1,p**3))
A145296_list = list(islice(A145296_gen(),20)) # Chai Wah Wu, May 04 2024