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

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

Author

Michel Lagneau, Oct 07 2017

Keywords

• nonn

Comments

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, ...

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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:

Mathematica

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 *)

PARI

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

Extensions

Edited by Robert Israel, Dec 28 2017