A330653 The prime numbers whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.

29, 41, 47, 61, 83, 101, 263, 281, 401, 463, 601, 607, 661, 809, 821, 863, 1129, 1303, 2063, 2267, 3121, 3181, 3301, 3343, 4001, 4603, 5309, 5581, 6007, 6043, 6803, 6863, 7129, 7309, 8009, 8681, 8821, 9721, 9967, 10903, 10909, 14143, 16903, 17209, 18521, 19421, 20063, 20201, 20407, 20807, 21143, 24281, 25147

Offset: 1

Author

Scott R. Shannon, Dec 22 2019

Keywords

• nonn
• base

Comments

This sequence lists the prime numbers whose digit sum A007953, concatenation of adjacent digit sums A053392, and concatenation of adjacent digit differences A040115, are also primes. Due to the digit sum being prime this is a subsequence of A046704.

For primes up to ten million there are 2268 entries, which is about one prime in every 293. The largest digit sum is 53 for a(1482) = 5986889, the largest adjacent digit sum concatenation is 171818141113 for a(2076) = 8999567, and the largest adjacent digit difference concatenation is 993247 for a(2099) = 9096481.

Examples

			a(1) = 29, as 2 + 9 = 11, '2 + 9' = 11, '|2 - 9|' = 7, and 29, 11, 7 are all prime.
a(7) = 263, as 2 + 6 + 3 = 11, '2 + 6' + '6 + 3' = 89, '|2 - 6|' + '|6 - 3|' = 43, and 263, 11, 89, 43 are all prime.
a(25) = 4001, as 4 + 0 + 0 + 1 = 5, '4 + 0' + '0 + 0' + '0 + 1' = 401, '|4 - 0|' + '|0 - 0|' + '|0 - 1|' = 401, and 4001, 5, 401 are all prime.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Chris K. Caldwell, How many primes are there?

Crossrefs

Cf. A000040, A007953, A040115, A053392, A046704.