A214342 Count of the decimal descendants of the n-th prime.

23, 22, 11, 23, 1, 14, 4, 40, 15, 6, 7, 13, 1, 14, 5, 0, 9, 16, 11, 4, 15, 1, 1, 0, 3, 10, 28, 0, 12, 0, 8, 1, 1, 9, 5, 1, 4, 1, 0, 2, 0, 6, 2, 5, 10, 19, 3, 5, 5, 6, 8, 5, 7, 0, 5, 3, 5, 8, 4, 1, 2, 5, 1, 2, 2, 0, 9, 5, 0, 7, 7, 2, 11, 9, 2, 2, 0, 0, 4, 28, 0, 7

Offset: 1

Author

Alex Ratushnyak, Jul 12 2012

Keywords

• nonn
• base

Comments

Prime q is a decimal descendant of prime p if q = p*10+k and 0<=k<=9.

The number of direct decimal descendants is A038800(p).

a(n) is the total count of direct decimal descendants of the n-th prime that are also prime, plus their decimal descendants that are prime, and so on.

Conjecture: no terms bigger than 35 after a(8)=40.

Examples

			prime(3)=5 has eleven descendants: 53, 59, 593, 599, 5939, 59393, 59399, 593933, 593993, 5939333, 59393339. So a(3)=11. All candidates of the form 5nnn1 and 5nnn7 are divisible by 3.
prime(5)=11, the only decimal descendant of 11 that is prime is 113, and because there are no primes between 1130 and 1140, a(5)=1.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A214280, A055781, A055782, A055783, A055784.

Programs

Maple

A214342 := proc(n)
    option remember;
    local a,p,k,d ;
    a := 0 ;
    p := ithprime(n) ;
    for k from 0 to 9 do
        d := 10*p+k ;
        if isprime(d) then
            a := a+1+procname(numtheory[pi](d)) ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jul 19 2012

Mathematica

Table[t = {Prime[n]}; cnt = 0; While[t = Select[Flatten[Table[10*i + {1, 3, 7, 9}, {i, t}]], PrimeQ]; t != {}, cnt = cnt + Length[t]]; cnt, {n, 100}] (* T. D. Noe, Jul 24 2012 *)