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

1,1

Links

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

Example

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.

Prog

(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<z, j++; p=prime(j); if(p%4==1, if(v[j]==0, t=0, print1(v[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

Crossrefs

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

Sequence in context: A158660 A358332 A158668 * A176635 A048422 A157651

Adjacent sequences: A145293 A145294 A145295 * A145297 A145298 A145299

Keyword

nonn

Author

Klaus Brockhaus, Oct 08 2008

Status

approved