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

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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.

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:

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 *)

Crossrefs

Cf. A215951.

Sequence in context: A176877 A356095 A323379 * A029563 A145665 A135806

Adjacent sequences: A215964 A215965 A215966 * A215968 A215969 A215970

Keyword

nonn

Author

Michel Lagneau, Aug 29 2012

Status

approved