A281707 - OEIS

%I #13 Aug 12 2023 04:51:36

%S 2,6,14,42,62,186,254,434,762,1302,1778,5334,7874,16382,23622,49146,

%T 55118,114674,165354,262142,344022,507842,786426,1048574,1523526,

%U 1834994,2080514,3145722,3554894,5504982,6241542,7340018,8126402,10664682,14563598,22020054

%N Even integers k such that phi(sum of even divisors of k) = sum of odd divisors of k.

%C The number of divisors of a(n) is a power of 2, and sum of even divisors = 2^(m+1), sum of odd divisors = 2^m for some m.

%C a(n) == 2, 6 (mod 8) or a(n) == 2, 6 (mod 12).

%C a(n) is of the form 2*p1*p2*...pk where p1, p2, ..., pk are Mersenne primes = 3, 7, 31, 127, 8191, ... (see A000668).

%e 62 is a term because its divisors are 1, 2, 31 and 62, the sum of the even divisors of 62 = 62 + 2 = 2^6, the sum of odd divisors = 1 + 31 = 2^5, and phi(2^6) = 2^5.

%p with(numtheory):

%p for n from 2 by 2 to 10^6 do:

%p x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:

%p    for k from 1 to n1 do:

%p     if irem(x[k],2)=0

%p      then

%p      s0:=s0+ x[k]:

%p      else

%p      s1:=s1+ x[k]:

%p     fi:

%p   od:

%p     if s1=phi(s0)

%p      then

%p      print(n):

%p      else

%p     fi:

%p od:

%t Select[2 * Range[10^6], (sodd = (s = DivisorSigma[1, #])/(2^(IntegerExponent[#, 2]+1) - 1)) == EulerPhi[s - sodd] &] (* _Amiram Eldar_, Aug 12 2023 *)

%o (PARI) isok(n) = eulerphi(sumdiv(n, d, d*((d % 2)==0))) == sumdiv(n, d, d*(d%2)); \\ _Michel Marcus_, Jan 28 2017

%Y Cf. A000010, A000593, A000668, A146076.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jan 28 2017

%E a(1) inserted by _Amiram Eldar_, Aug 12 2023