A110028 Primes with a prime number of digits, all of them prime, that add up to a prime.

23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257, 32323, 32327, 33223, 33353, 33377, 33533, 33773, 35227, 35353, 35533, 35537, 35573, 35753, 37223, 37337, 52237, 52253, 52727, 53353

Offset: 1

Author

Sergio Pimentel, Mar 31 2006

Keywords

• nonn
• base

Comments

First 7-digit number in this sequence is 2222333.

From Michael De Vlieger, Feb 02 2019: (Start)

First p-digit number in this sequence:

2 23

3 223

5 22573

7 2222333

11 22222222223

13 2222222225323

17 22222222222225237

19 2222222222222223527

23 22222222222222222232723

29 22222222222222222222222222577

31 2222222222222222222222222232257

37 2222222222222222222222222222222235773

...

(End)

Examples

			22573 is a term because 22573 is prime, it has five digits (5 is a prime), all digits (2,3,5,7) are prime, and the sum of the digits is 2+2+5+7+3 = 19, which is also a prime.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A062088.

Programs

Maple

a:=proc(n) local nn: nn:=convert(n,base,10): if isprime(n) and isprime(nops(nn)) and map(isprime,nn)=[seq(true,i=1..nops(nn))] and isprime(add(nn[j],j=1..nops(nn))) then n fi end: seq(a(k),k=1..60000); # Emeric Deutsch, Apr 02 2006

Mathematica

Select[Prime@ Range@ 6000, And[PrimeQ@ Length@ #, AllTrue[#, PrimeQ], PrimeQ@ Total@ #] &@ IntegerDigits@ # &] (* or *)
With[{p = {2, 3, 5, 7}}, Table[Select[FromDigits /@ Select[Tuples[p, {q}], PrimeQ@ Total@ # &], PrimeQ], {q, Prime@ Range@ 3}]] // Flatten (* Michael De Vlieger, Feb 02 2019 *)

Python

from sympy import isprime, nextprime
from itertools import islice, product
def agen(): # generator of terms
    p = 2
    while True:
        for d in product("2357", repeat=p-1):
            for last in "37":
                if isprime(sum(map(int, s:="".join(d) + last))):
                    if isprime(t:=int(s)):
                        yield t
        p = nextprime(p)
print(list(islice(agen(), 40))) # Michael S. Branicky, May 26 2023