Palindromic number: Difference between revisions

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A palindromic number is a symmentrical number written in some base a as a₁a₂a₃ ...|... a₃a₂a₁.

All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9: {11, 22, 33, 44, 55, 66, 77, 88, 99}. There are 90 palindromic numbers with three digits {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999} and also 90 palindromic numbers with four digits {1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999}, so there are 199 palindromic naumbers below 10⁴. Below 10⁵ there are 1099 palindromic numbers and for other exponents of 10ⁿ we have: 1999,10999,19999,109999,199999,1099999, ... (SIDN A070199). For some types of palindromic numbers these values are listed below in a table.

	10¹	10²	10³	10⁴	10⁵	10⁶	10⁷	10⁸	10⁹	10¹⁰
n natural	9	90		199	1099	1999	10999	19999	109999	199999
n even	5	9	49	89	489
n odd	5	10	60	110	610
n perfect square	3		6		13	14	19
n prime	4	5	20		113		781		5953
n squarefree	6	12	67	120	675
n non-square free (μ(n)=0)	3	6	41	78	423
n square with prime root	2		3		5
n with an even number of distinct prime factors (μ(n)=1)
n with an odd number of distinct prime factors (μ(n)=-1)

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-'''Palindromic number''' is a symmentrical [[number]] written in some base ''a'' as ''a''<sub>1</sub>''a''<sub>2</sub>''a''<sub>3</sub> ...|... ''a''<sub>3</sub>''a''<sub>2</sub>''a''<sub>1</sub>.
+A '''palindromic number''' is a symmentrical [[number]] written in some base ''a'' as ''a''<sub>1</sub>''a''<sub>2</sub>''a''<sub>3</sub> ...|... ''a''<sub>3</sub>''a''<sub>2</sub>''a''<sub>1</sub>.
-All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9: {11, 22, 33, 44, 55, 66, 77, 88, 99}. There are 90 palindromic numbers with three digits {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999} and also 90 palindromic numbers with four digits {1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999}, so there are 199 palindromic naumbers below 10<sup>4</sup>. Below 10<sup>5</sup> there are 1099 palindromic numbers and for other exponents of 10<sup>n</sup> we have: 1999,10999,19999,109999,199999,1099999, ... ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eismum.cgi SIDN A070199]). For some types of palindromic numbers these values are listed below in a table.
+All numbers in [[number system|base]] [[decimal|10]] with one [[digit]] {[[zero|0]], [[one|1]], [[two|2]], [[three|3]], 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9: {11, 22, 33, 44, 55, 66, 77, 88, 99}. There are 90 palindromic numbers with three digits {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999} and also 90 palindromic numbers with four digits {1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999}, so there are 199 palindromic naumbers below 10<sup>4</sup>. Below 10<sup>5</sup> there are 1099 palindromic numbers and for other exponents of 10<sup>n</sup> we have: 1999,10999,19999,109999,199999,1099999, ... ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eismum.cgi SIDN A070199]). For some types of palindromic numbers these values are listed below in a table.
 <table border="1" cellspacing="0" cellpadding="2">
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   </tr>
   <tr>
-    <td bgcolor="#FFCC99">''n'' with an even number of [[prime factor]]s
+    <td bgcolor="#FFCC99">''n'' with an even number of distinct [[prime factor]]s
       (&mu;(''n'')=1)</td>
     <td>&nbsp;</td>
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   </tr>
   <tr>
-    <td bgcolor="#FFCC99">''n'' with an odd number of prime factors
+    <td bgcolor="#FFCC99">''n'' with an odd number of distinct prime factors
       (&mu;(''n'')=-1)</td>
     <td>&nbsp;</td>