Consecutive Integers with Same Euler Phi Value
Jump to navigation
Jump to search
Theorem
Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function, defined on the strictly positive integers.
The equation:
- $\map \phi n = \map \phi {n + 1}$
is satisfied by integers in the sequence:
- $1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, \ldots$
This sequence is A001274 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Examples
$\phi$ of $1$ equals $\phi$ of $2$
- $\map \phi 1 = \map \phi 2 = 1$
$\phi$ of $3$ equals $\phi$ of $4$
- $\map \phi 3 = \map \phi 4 = 2$
$\phi$ of $15$ equals $\phi$ of $16$
- $\map \phi {15} = \map \phi {16} = 8$
$\phi$ of $104$ equals $\phi$ of $105$
- $\map \phi {104} = \map \phi {105} = 48$
$\phi$ of $164$ equals $\phi$ of $165$
- $\map \phi {164} = \map \phi {165} = 80$
$\phi$ of $194$ equals $\phi$ of $195$
- $\map \phi {194} = \map \phi {195} = 96$
$\phi$ of $255$ equals $\phi$ of $256$
- $\map \phi {255} = \map \phi {256} = 128$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $104$