A281819 Even numbers k such that half the sum of the even divisors equals the sum of the odd divisors and both are (the same) square.

2, 6, 162, 230, 238, 434, 530, 686, 690, 714, 770, 994, 1034, 1054, 1302, 1358, 1490, 1590, 1778, 1870, 2058, 2310, 2354, 2414, 2438, 2786, 2930, 2982, 3002, 3102, 3162, 3290, 3298, 3374, 3410, 3542, 3830, 4074, 4202, 4318, 4402, 4470, 4718, 4806, 5334, 5510, 5610, 5798, 5990, 6014, 6286

Offset: 1

Author

Michel Lagneau, Feb 03 2017

Keywords

• nonn

Comments

a(n) == 2 mod 4.

The corresponding squares are 1, 2^2, 11^2, 12^2, 12^2, 16^2, 18^2, 20^2, 24^2, 24^2, 24^2, 24^2, 24^2, 24^2, 32^2, 28^2, 30^2, 36^2, 32^2, 36^2, 40^2,...

There exists a subsequence {a(n)} intersection {A281707} = 6, 434, 1302, 1778, 7874, 23622, 114674, ... of numbers of the form 2p1*p2*...pk where p1, p2,...,pk are Mersenne primes = 3, 7, 31, 127, 8191,... (see A000668).

The corresponding squares are also powers of 2: 2^2, 2^8, 2^10, 2^10, 2^12,...

Examples

			162 is in the sequence because the divisors are {1, 2, 3, 6, 9, 18, 27, 54, 81, 162} => half sum of even divisors = (2 + 6 + 18 + 54 + 162)/2 = 11^2 and sum of odd divisors = 1 + 3 + 9 + 27 + 81 = 11^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000668, A281707, A006532.

Programs

Maple

with(numtheory):
for n from 2 by 2 to 10^5 do:
x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
   for k from 1 to n1 do:
    if irem(x[k],2)=0
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    s11:=sqrt(s1):s22:=sqrt(s0/2):
    if floor(s11)=s11 and floor(s22)=s22 and s11=s22
     then
     printf(`%d, `,n):
     else
    fi:
od:

PARI

forstep(k=1,1e3,2,if(issquare(sigma(k)), print1(2*k", "))) \\ Charles R Greathouse IV, Feb 06 2017

PARI

is(n)=n%4==2 && issquare(sigma(n/2)) \\ Charles R Greathouse IV, Feb 06 2017