A145215 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.

5, 41, 353, 1237, 2749, 3037, 10369, 6569, 27253, 38561, 14897, 33289, 27917, 171629, 143513, 76081, 37649, 373273, 399181, 63029, 133157, 637601, 425197, 94261, 499321, 910853, 229849, 149837

Offset: 1

Author

Vladimir Shevelev, Oct 05 2008

Keywords

• nonn

Comments

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.

Crossrefs

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

Programs

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))
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));
apply(f, S) \\ Charles R Greathouse IV, Feb 07 2013

Extensions

a(22) corrected by Charles R Greathouse IV, Feb 07 2013