A272328 Number of integers 1<=k<=n such that phi(n)=phi(n+k) where phi is Euler's totient function A000010.

1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 1, 0, 2, 1, 4, 3, 2, 0, 2, 2, 4, 0, 1, 1, 3, 3, 2, 2, 1, 0, 1, 4, 3, 3, 5, 1, 3, 1, 6, 2, 3, 0, 2, 2, 7, 0, 1, 1, 2, 1, 5, 6, 1, 0, 5, 5, 5, 0, 1, 0, 4, 0, 5, 5, 4, 0, 1, 4, 2, 4, 1, 3, 6, 4, 6, 3, 5, 2, 1, 3, 1, 5, 1, 1, 4, 1, 2

Offset

1,3

Comments

If n is odd, then phi(n) = phi(2n) so that a(n)>=1.

If n is a member of A043343, then a(n)=0.

It seems that every nonnegative integer appears in this sequence.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

For n=2: phi(2) = 1; whereas phi(2+1) = 2 and phi(2+2) = 2. Thus a(2) = 0.
For n=5: phi(5) = 4, phi(5+1)=2, phi(5+2)=6, phi(5+3) = 4, phi(5+4) = 6, and phi(5+5) = 4. Since phi(5) = phi(5+3) = phi(5+5), a(5) = 2.

Mathematica

Table[Count[Range@ n, k_ /; EulerPhi@ n == EulerPhi[n + k]], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

Prog

(Sage) [sum([1 for k in [1..n] if euler_phi(n)==euler_phi(n+k)]) for n in [1..1000]]
(Python)
from sympy import totient
nmax = 10**4
philist = [totient(i) for i in range(1, 2*nmax+1)]
A272328_list = [philist[i+1:2*(i+1)].count(philist[i]) for i in range(nmax)] # Chai Wah Wu, Apr 26 2016
(PARI) a(n) = my(x=eulerphi(n)); sum(k=1, n, eulerphi(n+k) == x); \\ Michel Marcus, Mar 08 2020

Crossrefs

Cf. A043343, A066659, A081373.

Sequence in context: A327274 A055378 A217921 * A334956 A335881 A239703

Adjacent sequences: A272325 A272326 A272327 * A272329 A272330 A272331

Keyword

nonn

Author

Tom Edgar, Apr 25 2016

Status

approved