Palindromes in Base 10 and Base 2

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Theorem

The following $n \in \Z$ are palindromic in both decimal and binary:

$0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$

This sequence is A007632 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

$n_{10}$ $n_2$
$0$ $0$
$1$ $1$
$3$ $11$
$5$ $101$
$7$ $111$
$9$ $1001$
$33$ $100 \, 001$
$99$ $1 \, 100 \, 011$
$313$ $100 \, 111 \, 001$
$585$ $1 \, 001 \, 001 \, 001$
$717$ $1 \, 011 \, 001 \, 101$
$7447$ $1 \, 110 \, 100 \, 010 \, 111$
$15 \, 351$ $11 \, 101 \, 111 \, 110 \, 111$
$32 \, 223$ $111 \, 110 \, 111 \, 011 \, 111$
$39 \, 993$ $1 \, 001 \, 110 \, 000 \, 111 \, 001$

$\blacksquare$


Sources