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

1,1

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,...

Links

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

Example

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.

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:

Prog

(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

Crossrefs

Cf. A000668, A281707, A006532.

Sequence in context: A051240 A003189 A323841 * A329980 A199482 A168649

Adjacent sequences: A281816 A281817 A281818 * A281820 A281821 A281822

Keyword

nonn

Author

Michel Lagneau, Feb 03 2017

Status

approved