OFFSET
1,2
COMMENTS
Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
A065338(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - Benedikt Otten, May 22 2013
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
REFERENCES
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
David Broadhurst and Wadim Zudilin, A magnetic double integral, arXiv:1708.02381 [math.NT], 2017. See p. 7
Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', N-movers: How much of the infinite board can I reach?
FORMULA
Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.
MAPLE
isA004613 := proc(n)
local p;
for p in numtheory[factorset](n) do
if modp(p, 4) <> 1 then
return false;
end if;
end do:
true;
end proc:
for n from 1 to 200 do
if isA004613(n) then
printf("%d, ", n) ;
end if;
end do: # R. J. Mathar, Nov 17 2014
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
select(q, [$1..500])[]; # Alois P. Heinz, Jan 13 2024
MATHEMATICA
ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500], Union[Mod[#, 4]&/@(FactorInteger[#][[All, 1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)
PROG
(PARI) for(n=1, 1000, if(sumdiv(n, d, isprime(d)*if((d-1)%4, 1, 0))==0, print1(n, ", ")))
(PARI) is(n)=n%4==1 && factorback(factor(n)[, 1]%4)==1 \\ Charles R Greathouse IV, Sep 19 2016
(Magma) [n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012
(Haskell)
a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved