A174024 List of primes of the form x^2+y^2 such that tau(x^2+y^2) = bigomega(x*y).

13, 17, 29, 37, 53, 101, 173, 197, 293, 677, 1373, 2213, 4493, 5333, 5477, 8837, 9413, 10613, 17957, 18773, 21317, 26573, 27893, 37253, 42437, 54293, 76733, 85853, 94253, 97973, 98597, 100493, 106277, 120413, 139133, 148997, 214373, 217157

Offset

1,1

Comments

bigomega(n) is the number of prime divisors of n (counted with multiplicity) (A001222) Because n = x^2+y^2 is prime, tau(n)= 2, and if we suppose x < y, then (x,y) = (2, p) with p prime or (x,y)=(1, 2q) with q prime.

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.

Links

Table of n, a(n) for n=1..38.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Example

13 = 2^2 + 3^2, bigomega(2*3) = 2.
17 = 1+4^2, bigomega(1*4) = 2.
994013 = 2^2 + 997^2, bigomega(2*997) = 2.

Maple

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ): k:=1:for x from 1 to 1000 do:for y from x to 1000 do:if tau(x^2+y^2)= bigomega(x*y) and type(x^2+y^2, prime)=true then T[k]:=x^2+y^2:k:=k+1:else fi:od :od:mini:=T[1]:ii:=1: for p from 1 to k-1 do:for n from 1 to k-1 do:if T[n]< mini then mini:= T[n]:ii:=n: indice:=U[n]: else fi:od:print(mini):T[ii]:= 99999999: ii:=1:mini:=T[1] :od:

Crossrefs

Cf. A020882, A002313, A001222, A001221 (primes counted without multiplicity), A046660, A144494.

Sequence in context: A154762 A079348 A349667 * A061381 A048520 A283407

Adjacent sequences: A174021 A174022 A174023 * A174025 A174026 A174027

Keyword

nonn

Author

Michel Lagneau, Mar 05 2010

Status

approved