Palindromic number

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, ... (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 + + + + +

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