Palindromic number (original) (raw)

A palindromic number is a symmetrical number written in some base a as _a_1_a_2_a_3 ...|... _a_3_a_2_a_1.

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, ... (SIDN A070199). 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 ann 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 | | | | | | | | | | |