A281707 Even integers k such that phi(sum of even divisors of k) = sum of odd divisors of k.

2, 6, 14, 42, 62, 186, 254, 434, 762, 1302, 1778, 5334, 7874, 16382, 23622, 49146, 55118, 114674, 165354, 262142, 344022, 507842, 786426, 1048574, 1523526, 1834994, 2080514, 3145722, 3554894, 5504982, 6241542, 7340018, 8126402, 10664682, 14563598, 22020054

Offset

1,1

Comments

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.

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

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

Links

Table of n, a(n) for n=1..36.

Example

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.

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 irem(x[k], 2)=0
     then
     s0:=s0+ x[k]:
     else
     s1:=s1+ x[k]:
    fi:
  od:
    if s1=phi(s0)
     then
     print(n):
     else
    fi:
od:

Mathematica

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

Prog

(PARI) isok(n) = eulerphi(sumdiv(n, d, d*((d % 2)==0))) == sumdiv(n, d, d*(d%2)); \\ Michel Marcus, Jan 28 2017

Crossrefs

Cf. A000010, A000593, A000668, A146076.

Sequence in context: A008325 A004066 A123383 * A225172 A212197 A151414

Adjacent sequences: A281704 A281705 A281706 * A281708 A281709 A281710

Keyword

nonn

Author

Michel Lagneau, Jan 28 2017

Extensions

a(1) inserted by Amiram Eldar, Aug 12 2023

Status

approved