A174053 Prime numbers of the form x^2+y^2 such that Moebius(x)* Moebius(y) = -1

5, 61, 109, 149, 157, 229, 269, 317, 389, 397, 461, 509, 557, 653, 701, 773, 797, 877, 941, 997, 1013, 1061, 1093, 1181, 1229, 1277, 1453, 1493, 1613, 1637, 1733, 1877, 1949, 1973, 1997, 2141, 2237, 2309, 2333, 2357, 2477, 2693, 2837, 2909, 2957, 3253

Offset

1,1

Comments

All terms == 5 (mod 8). - Robert Israel, May 06 2019

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

5 = 1^2 + 2^2 and Moebius(1)*Moebius(2) = (1) *(-1) = -1.
61 = 5^2 + 6^2 and Moebius(5)*Moebius(6) = (-1)*(1) = -1.

Maple

isA174053 := proc(n)
        local x, y ;
        if not isprime(n) then
                return false;
        end if;
        for x from 1 do
                if x^2 > n then
                        return false;
                end if;
                if issqr(n-x^2) then
                        y := sqrt(n-x^2) ;
                        if numtheory[mobius](x) * numtheory[mobius](y) = -1 then
                                return true;
                        end if;
                end if;
        end do:
end proc:
for n from 1 to 3000 do
        if isA174053(n) then
                printf("%d, \n", n) ;
        end if;
end do:
# R. J. Mathar, Jul 09 2012
# alternative
N:= 10^5: # to get all terms <= N
SF:= select(numtheory:-issqrfree, [$1..floor(sqrt(N))]):
Mp, Mm:= selectremove(numtheory:-mobius=1, SF):
R:= select(t -> t <= N and isprime(t), {seq(seq(s^2+t^2, s=Mp), t=Mm)}):
sort(convert(R, list)); # Robert Israel, May 06 2019

Mathematica

terms = 1000;
Reap[Do[p = x^2 + y^2; If[PrimeQ[p] && MoebiusMu[x] MoebiusMu[y] == -1, Sow[p]], {x, terms}, {y, x}]][[2, 1]] // Sort // Take[#, terms]& (* Jean-François Alcover, Aug 31 2020 *)

Crossrefs

Cf. A008683, A174049, A002313.

Sequence in context: A367337 A141967 A139915 * A107191 A321616 A182352

Adjacent sequences: A174050 A174051 A174052 * A174054 A174055 A174056

Keyword

nonn

Author

Michel Lagneau, Mar 06 2010

Status

approved