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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.
1
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 03 2017
STATUS
approved