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A004613 Numbers that are divisible only by primes congruent to 1 mod 4. 23
1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421 (list; graph; refs; listen; history; text; internal format)
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

Product(A079260(A027748(a(n),k)): k=1..A001221(a(n))) = 1. - Reinhard Zumkeller, Jan 07 2013

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 n such that there is a "knight's move" of Euclidean distance sqrt(n) 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.

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

'trichoplax', pseudonymous Stack Exchange user, 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

MATHEMATICA

ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-Fran├žois 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

Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.

Cf. A004614.

Sequence in context: A087445 A020882 A081804 * A008846 A162597 A120960

Adjacent sequences:  A004610 A004611 A004612 * A004614 A004615 A004616

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 22 16:24 EDT 2019. Contains 325224 sequences. (Running on oeis4.)