A346979 Count of the prime decimal descendants of n.

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset: 0

Author

Ya-Ping Lu, Aug 09 2021

Keywords

• nonn
• base

Comments

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.

a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).

a(A032352(m)) = 0; a(A119289(m)) = 0.

A214342 is a subset, as A214342(m) = a(prime(m)).

Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

Examples

			a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339
		

Crossrefs

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Programs

Mathematica

Table[Length@Rest@Flatten[FixedPointList[(b=#;Select[Flatten[(a=#;FromDigits/@(Join[IntegerDigits@a,{#}]&/@If[b=={0},Range@9,{1,3,7,9}]))&/@b],PrimeQ])&,{n}]],{n,0,76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

Python

from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))