A145215 - OEIS

%I #9 Dec 28 2023 19:39:09

%S 5,41,353,1237,2749,3037,10369,6569,27253,38561,14897,33289,27917,

%T 171629,143513,76081,37649,373273,399181,63029,133157,637601,425197,

%U 94261,499321,910853,229849,149837

%N a(n) is the minimal prime of the form 4k+1 for which s=A008784(n) is the minimal positive integer such that s*a(n)-floor(sqrt(s*a(n)))^2 is a square.

%C See the conjecture in the comment at A145047. In addition, I conjecture that for every such s there exist infinitely many primes of the form 4k+1.

%o (PARI) f(s)=forprime(p=2,,if(p%4>1 || !issquare(s*p-sqrtint(s*p)^2),next);for(i=1,s-1,if(issquare(i*p-sqrtint(i*p)^2), next(2)));return(p))

%o S=select(n->if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1, vector(150,i,i));

%o apply(f, S) \\ _Charles R Greathouse IV_, Feb 07 2013

%Y Cf. A145016, A145022, A145023, A145047-A145050, A008487.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Oct 05 2008

%E a(22) corrected by _Charles R Greathouse IV_, Feb 07 2013