A282515 Numbers m such that phi(sum of the divisors of m) = phi(sum of the distinct prime divisors of m) where phi is the Euler totient function.

3, 6, 10, 22, 34, 142, 178, 214, 382, 862, 1402, 2302, 5182, 9098, 15398, 17398, 21178, 23602, 279934, 289558, 296734, 368062, 900754, 944782, 1079374, 1563442, 1572862, 1990654, 2116342, 2505886, 2584882, 2691574, 2858698, 2883058, 3351214, 3909046

Offset: 1

Author

Michel Lagneau, Feb 17 2017

Keywords

• nonn

Comments

Or numbers m such that A000010(A000203(m)) = A000010(A008472(m)).

For n > 1, we observe that a(n) is semiprime of the form a(n) = 2p with p = 3, 5, 11, 17, 71, 89, 107, 191, 431, 701, 1151, 2591, 4549, 7699, 8699, 10589, 11801, ... Except for the primes 3, 4549 and 7699 in the first 35 terms (from 6 until 3909046), the primes p are of the form 6k - 1.

Examples

			34 is in the sequence because A000010(A000203(34)) = A000010(54) = 18 and A000010(A008472(34)) = A000010(19) = 18.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..289 (terms below 10^10)

Crossrefs

Cf. A000010, A000203, A008472, A062401.

Programs

Maple

with(numtheory):
for n from 2 to 200000 do:
x:=divisors(n):n0:=nops(x):y:=factorset(n):n1:=nops(y):
   s0:=sum(‘x[i]’, ‘i’=1..n0):s1:=sum(‘y[i]’, ‘i’=1..n1):
    if phi(s1)=phi(s0)
     then
       print(n):
       else
     fi:
od:

Mathematica

Select[Range[10^6], EulerPhi@ DivisorSigma[1, #] == EulerPhi[Total@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Feb 17 2017 *)

PARI

isok(n) = my(f=factor(n)); eulerphi(sigma(n)) == eulerphi(vecsum(f[,1])); \\ Michel Marcus, Feb 25 2017