A061783 Luhn primes: primes p such that p + (p reversed) is also a prime.

229, 239, 241, 257, 269, 271, 277, 281, 439, 443, 463, 467, 479, 499, 613, 641, 653, 661, 673, 677, 683, 691, 811, 823, 839, 863, 881, 20011, 20029, 20047, 20051, 20101, 20161, 20201, 20249, 20269, 20347, 20389, 20399, 20441, 20477, 20479, 20507

Offset: 1

Author

Amarnath Murthy, May 24 2001

Keywords

• nonn
• base
• easy

Comments

a(n) has an odd number of digits, as otherwise a(n) + reverse(a(n)) is a multiple of 11. For a(n) > 10, a(n) is prime and thus odd, and therefore the first digit of a(n) is even as otherwise a(n) + reverse(a(n)) is even and composite. - Chai Wah Wu, Aug 19 2015

See A072385 for the resulting primes p + reverse(p) = A056964(p). - M. F. Hasler, Sep 26 2019

Named by Cira and Smarandache (2014) after Norman Luhn, who noted the property of the prime 229 on the Prime Curios! website. - Amiram Eldar, Jun 05 2021

Examples

			229 is a term since 229 is a prime and so is 229 + 922 = 1151.
		

Links

• Harry J. Smith and Chai Wah Wu, Table of n, a(n) for n = 1..50598, giving all terms below 9*10^6 (The first 1000 terms from Harry J. Smith)
• Octavian Cira and Florian Smarandache, Luhn prime numbers, Theory and Applications of Mathematics & Computer Science, Vol. 5, No. 1 (2015), pp. 29-36; preprint, 2014.
• G. L. Honaker, Jr. and Chris Caldwell, eds., 229, Prime Curios!, November 19, 2001.
• Chai Wah Wu, 3010506 terms, 11MB zipped file of all terms below 10^9.

Crossrefs

Cf. A004086 (reverse), A004087 (primes reversed), A056964 (reverse & add), A072385, A086002 (similar, using "rotate" instead of "reverse").

Programs

Magma

[NthPrime(n): n in [1..2400] | IsPrime(s) where s is NthPrime(n)+Seqint(Reverse(Intseq(NthPrime(n))))]; // Bruno Berselli, Aug 05 2013

Mathematica

Select[Prime[Range[3000]],PrimeQ[#+FromDigits[Reverse[IntegerDigits[#]]]]&] (* Harvey P. Dale, Nov 27 2010 *)

PARI

isok(p) = { isprime(p) && isprime(p + fromdigits(Vecrev(digits(p)))) } \\ Harry J. Smith, Jul 28 2009

PARI

select( is_A061783(p)=isprime(A056964(p)) && isprime(p), primes(8713)) \\  A056964(p)=p+fromdigits(Vecrev(digits(p))). There is no term with 4 digits or starting with an odd digit, i.e., no candidate between prime(168) = 997 and prime(2263) = 20011. Using primes up to prime(8713) = 89989 ensures the list of 5-digit terms is complete. - M. F. Hasler, Sep 26 2019

Python

from sympy import isprime, prime
A061783 = [prime(n) for n in range(1,10**5) if isprime(prime(n)+int(str(prime(n))[::-1]))] # Chai Wah Wu, Aug 14 2014

Extensions

Corrected and extended by Patrick De Geest, May 26 2001

Cross-references added by M. F. Hasler, Sep 26 2019