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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.

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proposed

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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 full square.

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

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

5, 41, 353, 1237, 2749, 3037, 10369, 6569, 27253, 38561, 14897, 33289, 27917, 171629, 143513, 76081, 37649, 373273, 399181, 63029, 133157, 657601, 637601, 425197, 94261, 499321, 910853, 229849, 149837

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

(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

Cf. A145016 , A145022 , A145023 , A145047 A148048 A145049 -A145050 , A008487.

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

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editing

_Vladimir Shevelev (shevelev(AT)bgu.ac.il), _, Oct 05 2008

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

5, 41, 353, 1237, 2749, 3037, 10369, 6569, 27253, 38561, 14897, 33289, 27917, 171629, 143513, 76081, 37649, 373273, 399181, 63029, 133157, 657601, 425197, 94261, 499321, 910853, 229849, 149837

1,1

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

A145016 A145022 A145023 A145047 A148048 A145049 A145050 A008487

nonn

Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 05 2008

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