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

Author

Michel Lagneau, Mar 05 2010

Keywords

• nonn

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.

Examples

			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.
		

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

• 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.

Crossrefs

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

Programs

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: