A281707 -id:A281707 - cp's OEIS frontend

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

Michel Lagneau, Feb 03 2017

nonn

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,...

			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.

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

Cf. A000668, A281707, A006532.

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:

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

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

A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.

Original entry on oeis.org

2, 20, 40, 48, 68, 176, 212, 304, 328, 944, 1360, 1712, 1888, 2320, 2344, 2864, 4240, 7120, 7888, 7984, 8448, 8960, 11920, 12032, 14416, 14592, 15536, 17492, 20224, 21520, 23984, 24208, 24592, 25904, 26112, 28160, 29440, 30464, 34560, 35920, 36352, 40528, 41296

Offset: 1

Michel Lagneau, Oct 07 2017

nonn

Or even integers k such that A002322(A146076(k)) = A000593(k).
Observations:
The primes a(n)/4: {5, 17, 53, 4373, 13121, ...} are of the form 2*3^m - 1, m > 0 (A079363).
The primes a(n)/8: {5, 41, 293, 4941257, ...} are of the form 6*7^m - 1, m = 0, 1, ... (primes in A198688).
The set of the primes {a(n)/16} = {3, 11, 19, 59, 107, 179, 499, 971, 1499, 1619, ...} contains the primes of the form 4*3^(2m+1) - 1 = {11, 107, 971, ...}, m = 0, 1, ...

			68 is in the sequence because A002322(A146076(68)) = A002322(108) = 18 and A000593(68) = 18.

Robert Israel, Table of n, a(n) for n = 1..738

Cf. A000593, A000668, A002322, A146076, A281707.

with(numtheory):
for n from 2 by 2 to 10^6 do:
x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
   for k from 1 to n1 do:
    if type(x[k],even)
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    if s1=lambda(s0)
     then
     printf(`%d, `,n):
     else
    fi:
od:

fQ[n_] :=
Block[{d = Divisors@n},
  CarmichaelLambda[Plus @@ Select[d, EvenQ]] ==
Plus @@ Select[d, OddQ]]; Select[2 Range@2000, fQ] (* Robert G. Wilson v, Oct 07 2017 *)

is(n)=if(n%2, return(0)); my(s=valuation(n,2),d=sigma(n>>s)); lcm(znstar(d*(2^(s+1)-2))[2])==d \\ Charles R Greathouse IV, Dec 26 2017

Edited by Robert Israel, Dec 28 2017

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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.

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A293356 Even integers k such that lambda(sum of even divisors of k) = sum of odd divisors of k.

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