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.
165, 330, 429, 495, 660, 741, 805, 825, 858, 990, 1045, 1155, 1173, 1235, 1245, 1287, 1309, 1320, 1482, 1485, 1610, 1645, 1650, 1716, 1815, 1955, 1980, 2090, 2145, 2223, 2261, 2301, 2310, 2346, 2365, 2470, 2475, 2490, 2574, 2618, 2635, 2640, 2765, 2795, 2821
Offset: 1
Keywords
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Cf. A215951.
Programs
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Maple
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:
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Mathematica
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 *)