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, ... (sequence A070199 in the 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 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

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

• Palindromic prime
• Palindrome

External links

• Palindromic Numbers to 100,000 from Ask Dr. Math