Palindromic Number - Decimal Palindromic Numbers

Decimal Palindromic Numbers

All numbers in base 10 with one digit are palindromic. 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 (Using the Rule of product: 9 choices for the first digit - which determines the third digit as well - multiplied by 10 choices for the second digit):

{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: (Again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two)

{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, … (sequence A070199 in 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 10 19 109 199 1099 1999 10999 19999 109999 199999
n even 5 9 49 89 489 889 4889 8889 48889 88889
n odd 5 10 60 110 610 1110 6110 11110 61110 111110
n square 4 7 14 15 20 31
n cube 3 4 5 7 8
n prime 4 5 20 113 781 5953
n squarefree 6 12 67 120 675 1200 6821 12160 + +
n non-squarefree (μ(n)=0) 4 7 42 79 424 799 4178 7839 + +
n square with prime root 2 3 5
n with an even number of distinct prime factors (μ(n)=1) 2 6 35 56 324 583 3383 6093 + +
n with an odd number of distinct prime factors (μ(n)=-1) 4 6 32 64 351 617 3438 6067 + +
n even with an odd number of prime factors 1 2 9 21 100 180 1010 6067 + +
n even with an odd number of distinct prime factors 3 4 21 49 268 482 2486 4452 + +
n odd with an odd number of prime factors 3 4 23 43 251 437 2428 4315 + +
n odd with an odd number of distinct prime factors 4 5 28 56 317 566 3070 5607 + +
n even squarefree with an even number of (distinct) prime factors 1 2 11 15 98 171 991 1782 + +
n odd squarefree with an even number of (distinct) prime factors 1 4 24 41 226 412 2392 4221 + +
n odd with exactly 2 prime factors 1 4 25 39 205 303 1768 2403 + +
n even with exactly 2 prime factors 2 3 11 64 413 + +
n even with exactly 3 prime factors 1 3 14 24 122 179 1056 1400 + +
n even with exactly 3 distinct prime factors 0 1 18 44 250 390 2001 2814 + +
n odd with exactly 3 prime factors 0 1 12 34 173 348 1762 3292 + +
n Carmichael number 0 0 0 0 0 1 1 1 1 1
n for which σ(n) is palindromic 6 10 47 114 688 1417 5683 + + +

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