A243442 Primes p such that, in base 2, p - digitsum(p) is also a prime.

5, 23, 71, 83, 101, 113, 197, 281, 317, 353, 359, 373, 401, 467, 599, 619, 683, 739, 751, 773, 977, 1091, 1097, 1103, 1217, 1223, 1229, 1237, 1283, 1303, 1307, 1429, 1433, 1489, 1553, 1559, 1601, 1607, 1613, 1619, 1699, 1873, 1879, 2039, 2347, 2357, 2389

Offset: 1

Author

Anthony Sand, Jun 05 2014

Keywords

• nonn
• base

Comments

In all bases b, x = n - digitsum(n) is always divisible by b-1, therefore x can be prime only in base 2 and bases b for which b-1 is prime. For example, in base 10, n - digitsum(n) is always divisible by 10 - 1 = 9 -- see A066568 and A068395. In base 8, 9 = 11, therefore 11 - digitsum(11) = 9 - 2 = 7 is divisible by 7.

Examples

			5 - digitsum(5,base=2) = 5 - digitsum(101) = 5 - 2 = 3.
23 - digitsum(10111) = 23 - 4 = 19.
71 - digitsum(1000111) = 71 - 4 = 67.
83 - digitsum(1010011) = 83 - 4 = 79.
101 - digitsum(1100101) = 101 - 4 = 97.
		

Links

• Anthony Sand, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A243441.

Programs

Mathematica

Select[Prime[Range[400]],PrimeQ[#-Total[IntegerDigits[#,2]]]&] (* Harvey P. Dale, May 15 2019 *)

PARI

isok(n) = isprime(n) && isprime(n - hammingweight(n)); \\ Michel Marcus, Jun 05 2014