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A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

%I A004613 #75 May 15 2025 19:31:36
%S A004613 1,5,13,17,25,29,37,41,53,61,65,73,85,89,97,101,109,113,125,137,145,
%T A004613 149,157,169,173,181,185,193,197,205,221,229,233,241,257,265,269,277,
%U A004613 281,289,293,305,313,317,325,337,349,353,365,373,377,389,397,401,409,421
%N A004613 Numbers that are divisible only by primes congruent to 1 mod 4.
%C A004613 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
%C A004613 A065338(a(n)) = 1. - _Reinhard Zumkeller_, Jul 10 2010
%C A004613 Product_{k=1..A001221(a(n))} A079260(A027748(a(n),k)) = 1. - _Reinhard Zumkeller_, Jan 07 2013
%C A004613 A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - _Benedikt Otten_, May 22 2013
%C A004613 Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - _Michael Somos_, Jul 26 2013
%C A004613 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.
%C A004613 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
%C A004613 Sequence closed under multiplication. Odd values of A031396 and their powers. These are the only numbers m that satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1. - _Klaus Purath_, May 12 2025
%D A004613 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
%H A004613 T. D. Noe, <a href="/A004613/b004613.txt">Table of n, a(n) for n = 1..10000</a>
%H A004613 David Broadhurst and Wadim Zudilin, <a href="https://arxiv.org/abs/1708.02381">A magnetic double integral</a>, arXiv:1708.02381 [math.NT], 2017. See p. 7
%H A004613 Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', <a href="https://codegolf.stackexchange.com/questions/179747/n-movers-how-much-of-the-infinite-board-can-i-reach">N-movers: How much of the infinite board can I reach?</a>
%F A004613 Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.
%p A004613 isA004613 := proc(n)
%p A004613     local p;
%p A004613     for p in numtheory[factorset](n) do
%p A004613         if modp(p,4) <> 1 then
%p A004613             return false;
%p A004613         end if;
%p A004613     end do:
%p A004613     true;
%p A004613 end proc:
%p A004613 for n from 1 to 200 do
%p A004613     if isA004613(n) then
%p A004613         printf("%d,",n) ;
%p A004613     end if;
%p A004613 end do: # _R. J. Mathar_, Nov 17 2014
%p A004613 # second Maple program:
%p A004613 q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
%p A004613 select(q, [$1..500])[];  # _Alois P. Heinz_, Jan 13 2024
%t A004613 ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* _Jean-François Alcover_, May 05 2011 *)
%t A004613 Select[Range[500],Union[Mod[#,4]&/@(FactorInteger[#][[All,1]])]=={1}&] (* _Harvey P. Dale_, Mar 08 2017 *)
%o A004613 (PARI) for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-1)%4,1,0))==0,print1(n,",")))
%o A004613 (PARI) is(n)=n%4==1 && factorback(factor(n)[,1]%4)==1 \\ _Charles R Greathouse IV_, Sep 19 2016
%o A004613 (Magma) [n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // _Vincenzo Librandi_, Aug 21 2012
%o A004613 (Haskell)
%o A004613 a004613 n = a004613_list !! (n-1)
%o A004613 a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
%o A004613 -- _Reinhard Zumkeller_, Jan 07 2013
%Y A004613 Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.
%Y A004613 Cf. A004614.
%K A004613 nonn,nice,easy
%O A004613 1,2
%A A004613 _N. J. A. Sloane_