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A174024 List of primes of the form x^2+y^2 such that tau(x^2+y^2) = bigomega(x*y) 0


%S 13,17,29,37,53,101,173,197,293,677,1373,2213,4493,5333,5477,8837,

%T 9413,10613,17957,18773,21317,26573,27893,37253,42437,54293,76733,

%U 85853,94253,97973,98597,100493,106277,120413,139133,148997,214373,217157

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

%C 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.

%D 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.

%D 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.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.

%e 13 = 2^2 + 3^2, bigomega(2*3) = 2.

%e 17 = 1+4^2, bigomega(1*4) = 2.

%e 994013 = 2^2 + 997^2, bigomega(2*997) = 2.

%p 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:

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

%K nonn

%O 1,1

%A _Michel Lagneau_, Mar 05 2010

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