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A174024
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List of primes of the form x^2+y^2 such that tau(x^2+y^2) = bigomega(x*y)
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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].
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EXAMPLE
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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.
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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