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A145017 Squarefree positive integers k for which k-(floor(sqrt(k)))^2 is a perfect square. 3


%S 1,2,5,10,13,17,26,29,34,37,53,58,65,73,82,85,97,101,109,122,130,137,

%T 145,170,173,178,185,194,197,205,221,226,229,241,257,265,281,290,293,

%U 298,305,314,349,362,365,370,377,386,397,401,409,442,445,457,466,485,493

%N Squarefree positive integers k for which k-(floor(sqrt(k)))^2 is a perfect square.

%C If an odd prime p divides a(n) then it has the form 4k+1.

%C Conjecture. For every n>=1 there exist infinitely many primes p of the form 4k+1 for which for a(n) > 1 we have s*p-(floor(sqrt(s*p)))^2 is not a perfect square for s=1,...,a(n)-1 while a(n)*p-(floor(sqrt(a(n)p))^2 is a perfect square. (See A145016(s=1) and A145022, A145023, A145047, A145048, A145149, A145050 correspondingly for s=2, s=5, s=10, s=13, s=17, s=26.) - _Vladimir Shevelev_, Sep 30 2008

%H Andrew Howroyd, <a href="/A145017/b145017.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Range@ 500, And[SquareFreeQ@ #, IntegerQ@ Sqrt[# - Floor[Sqrt@ #]^2]] &] (* _Michael De Vlieger_, Jan 12 2020 *)

%o (PARI) is(n)={issquarefree(n) && issquare(n-sqrtint(n)^2)} \\ _Andrew Howroyd_, Jan 12 2020

%Y Cf. A005117, A020893, A145016.

%Y Cf. A145016, A145022, A045023, A145047, A145048, A145049, A145050.

%K nonn

%O 1,2

%A _Vladimir Shevelev_, Sep 29 2008

%E Missing a(40) inserted and terms a(42) and beyond from _Andrew Howroyd_, Jan 12 2020

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Last modified June 19 00:37 EDT 2021. Contains 345125 sequences. (Running on oeis4.)