Consecutive Integers with Same Euler Phi Value

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Theorem

Let $\phi: \Z_{>0} \to \Z_{>0}$ be the Euler $\phi$ function, defined on the strictly positive integers.

The equation:

$\phi \left({n}\right) = \phi \left({n + 1}\right)$

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 $4$

$\phi \left({3}\right) = \phi \left({4}\right) = 2$


$\phi$ of $15$ equals $\phi$ of $16$

$\phi \left({15}\right) = \phi \left({16}\right) = 8$


$\phi$ of $104$ equals $\phi$ of $105$

$\sigma \left({104}\right) = \sigma \left({105}\right) = 48$


$\phi$ of $164$ equals $\phi$ of $165$

$\sigma \left({164}\right) = \sigma \left({165}\right) = 80$


$\phi$ of $194$ equals $\phi$ of $195$

$\sigma \left({194}\right) = \sigma \left({195}\right) = 96$


$\phi$ of $255$ equals $\phi$ of $256$

$\sigma \left({255}\right) = \sigma \left({256}\right) = 128$


Sources