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A215967 Numbers n such that the absolute value of the difference between the sum of the distinct prime divisors of n that are congruent to 1 mod 4 and the sum of the distinct prime divisors of n that are congruent to 3 mod 4 is a square.

%I A215967 #12 Sep 09 2019 03:28:06
%S A215967 165,330,429,495,660,741,805,825,858,990,1045,1155,1173,1235,1245,
%T A215967 1287,1309,1320,1482,1485,1610,1645,1650,1716,1815,1955,1980,2090,
%U A215967 2145,2223,2261,2301,2310,2346,2365,2470,2475,2490,2574,2618,2635,2640,2765,2795,2821
%N A215967 Numbers n such that the absolute value of the difference between the sum of the distinct prime divisors of n that are congruent to 1 mod 4 and the sum of the distinct prime divisors of n that are congruent to 3 mod 4 is a square.
%H A215967 Amiram Eldar, <a href="/A215967/b215967.txt">Table of n, a(n) for n = 1..10000</a>
%e A215967 2365 is in the sequence because 2365 = 5*11*43 and (11+43) - 5 = 49 is a square, where {11, 43} == 3 mod 4 and 5 ==1 mod 4.
%p A215967 with(numtheory):for n from 2  to 1000 do:x:=factorset(n):n1:=nops(x):s1:=0:s3:=0:for m from 1 to n1 do: if irem(x[m], 4)=1 then s1:=s1+x[m]:else if irem(x[m], 4)=3 then s3:=s3+x[m]:else fi:fi:od:x:=abs(s1-s3):y:=sqrt(x):if s1>0 and s3>0 and y=floor(y) then printf(`%d, `, n):else fi:od:
%t A215967 aQ[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, (t1 = Total[Select[p, Mod[#, 4] == 1 &]]) > 0 && (t2 = Total[Select[p, Mod[#, 4] == 3 &]]) > 0 && IntegerQ@Sqrt@Abs[t1 - t2]]; Select[Range[3000], aQ] (* _Amiram Eldar_, Sep 09 2019 *)
%Y A215967 Cf. A215951.
%K A215967 nonn
%O A215967 1,1
%A A215967 _Michel Lagneau_, Aug 29 2012