A069217 Numbers n such that phi(n) + sigma(n) = n + reversal(n).

1, 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181

Offset: 1

Author

Joseph L. Pe, Apr 11 2002

Keywords

• base
• nonn

Comments

Note that all terms so far are palindromes.

It is obvious that if n is a term of the sequence greater than 1 then n is prime iff n is a palindrome. Do there exist composite terms in the sequence? - Farideh Firoozbakht, Jan 28 2006 Answer: Yes, see next comment.

Giovanni Resta writes (Sep 06 2006): The smallest composite number such that n+rev(n)=phi(n)+sigma(n) is n = 3197267223 = 3 * 79 * 677 * 19927 with rev(n) = 3227627913, phi(n) = 2101316256, sigma(n) = 4323578880 and 3197267223+3227627913 = 6424895136 = 2101316256+4323578880.

Examples

			phi(101) + sigma(101) = 202 = 101 + 101 = 101 + reversal(101).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..700

Crossrefs

Contains composite terms, so is strictly different from A002385.

Programs

Mathematica

Select[Range[5*10^4], EulerPhi[ # ] + DivisorSigma[1, # ] == # + FromDigits[Reverse[IntegerDigits[ # ]]] &]

Formula

If p is prime and rev(p)=p then p+rev(p)=2p=phi(p)+sigma(p) so all palindromic primes are in the sequence. - Farideh Firoozbakht, Sep 12 2006