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

%I A145296 #14 May 04 2024 02:55:02
%S A145296 57,239,1985,10133,9466,11389,27590,51412,153765,344464,107551,296344,
%T A145296 172078,432436,931837,753090,676541,2321221,2027724,3394758,1706203,
%U A145296 4841182,1438398,2947125,398366,5657795,4942017,9400802,11906503
%N A145296 Smallest k such that k^2 + 1 is divisible by A002144(n)^3.
%H A145296 Chai Wah Wu, <a href="/A145296/b145296.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..150 from Klaus Brockhaus)
%e A145296 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.
%o A145296 (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], ","))))}
%o A145296 (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
%o A145296 (Python)
%o A145296 from itertools import islice
%o A145296 from sympy import nextprime, sqrt_mod_iter
%o A145296 def A145296_gen(): # generator of terms
%o A145296     p = 1
%o A145296     while (p:=nextprime(p)):
%o A145296         if p&3==1:
%o A145296             yield min(sqrt_mod_iter(-1,p**3))
%o A145296 A145296_list = list(islice(A145296_gen(),20)) # _Chai Wah Wu_, May 04 2024
%Y A145296 Cf. A002144 (primes of form 4n+1), A002313 (-1 is a square mod p), A059321, A145297, A145298, A145299.
%K A145296 nonn
%O A145296 1,1
%A A145296 _Klaus Brockhaus_, Oct 08 2008