A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

2, 574, 3346, 12474, 19598, 19710, 42770, 73062, 93310, 133630, 250510, 365330, 425898, 485758, 546530, 761022, 782690, 1254430, 1460290, 1628926, 2139790, 2174018, 2286954, 2332798, 2845154, 3185870, 3630146, 4562510, 5089394, 5444010, 5656770, 6265870, 6377618

Offset: 1

Author

Michel Lagneau, Jan 31 2025

Keywords

• nonn

Comments

Let s2 = 4*A001157(m/2) be the sum of the squares of the even divisors of m and s1 = A050999(m) be the sum of the squares of the odd divisors of m. We observe that s2/s1 = 4.

Examples

			574 is in the sequence because: its divisors are {1, 2, 7, 14, 41, 82, 287, 574}; the sum of squares of the odd divisors is 84100 which is square, and the sum of squares of the even divisors is 336400 which is square.
		

Crossrefs

Cf. A000203, A000593, A001157, A050999, A146076.

Programs

Maple

with(numtheory):nn:=10^7:
for m from 2 by 2 to nn do:
 d:=divisors(m):
   s1:=0: s2:=0:
   for i in d do
    if i::odd then s1:=s1+i^2 else s2:=s2+i^2 fi
   od:
   if issqr(s2) and issqr(s1) then print(m) fi
 od:

Mathematica

Select[Range[2,10^6,2],AllTrue[{Sqrt[Total[Select[Divisors[#],OddQ]^2]],Sqrt[Total[Select[Divisors[#],EvenQ]^2]]},IntegerQ]&] (* James C. McMahon, Feb 10 2025 *)

PARI

isok(k) = !(k%2) && issquare(sumdiv(k, d, if (d%2, d^2))) && issquare(sumdiv(k, d, if (1-d%2, d^2))); \\ Michel Marcus, Feb 22 2025