Consecutive Integers with Same Euler Phi Value

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): $104$