Palindromic number
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A palindromic number is a symmetrical 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 numbers below 104. Below 105 there are 1099 palindromic numbers and for other exponents of 10n we have: 1999,10999,19999,109999,199999,1099999, ... Template:OEIS. For some types of palindromic numbers these values are listed below in a table. Here 0 is included.
| 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 square-free | 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) | 2 | 6 | 35 | 56 | 324 | + | + | + | + | + |
| n with an odd number of distinct prime factors (μ(n)=-1) | 5 | 7 | 33 | 65 | 352 | + | + | + | + | + |
| n even with an odd number of prime factors | ||||||||||
| n even with an odd number of distinct prime factors | 1 | 2 | 9 | 21 | 100 | + | + | + | + | + |
| n odd with an odd number of prime factors | 0 | 1 | 12 | 37 | 204 | + | + | + | + | + |
| n odd with an odd number of distinct prime factors | 0 | 0 | 4 | 24 | 139 | + | + | + | + | + |
| n even squarefree with an even number of distinct prime factors | 1 | 2 | 11 | 15 | 98 | + | + | + | + | + |
| n odd squarefree with an even number of distinct prime factors | 1 | 4 | 24 | 41 | 226 | + | + | + | + | + |
| n odd with exactly 2 prime factors | 1 | 4 | 25 | 39 | 205 | + | + | + | + | + |
| n even with exactly 2 prime factors | 2 | 3 | 11 | 64 | + | + | + | + | + | |
| n even with exactly 3 prime factors | 1 | 3 | 14 | 24 | 122 | + | + | + | + | + |
| n even with exactly 3 distinct prime factors | ||||||||||
| n odd with exactly 3 prime factors | 0 | 1 | 12 | 34 | 173 | + | + | + | + | + |
| n Carmichael number | 0 | 0 | 0 | 0 | 0 | 1+ | + | + | + | + |
| n for which σ(n) is palindromic | 6 | 10 | 47 | 114 | 688 | + | + | + | + | + |
| add more | ||||||||||
Buckminster Fuller referred to palindromic numbers as Scheherezade numbers in his book "Synergetics", since Scheherezade was the name of the story telling wife in the "1001 Arabian Nights"
See also
External links
- Palindromic Numbers to 100,000 (http://mathforum.org/library/drmath/view/57170.html) from Ask Dr. Math
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