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A346979 Count of the prime decimal descendants of n.

%I A346979 #12 Oct 03 2021 19:21:05
%S A346979 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,
%T A346979 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,
%U A346979 6,19,2,9,25,16,0,7,4,4,3,11,2,7,7,4,1,15,2,8,8
%N A346979 Count of the prime decimal descendants of n.
%C A346979 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.
%C A346979 a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).
%C A346979 a(A032352(m)) = 0; a(A119289(m)) = 0.
%C A346979 A214342 is a subset, as A214342(m) = a(prime(m)).
%C A346979 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.
%e A346979 a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
%e A346979        _____ 4__________________________
%e A346979       /      |                          \
%e A346979      41   ___43______________            47
%e A346979     /    /   |               \             \
%e A346979   419  431  433               439          479
%e A346979             / \              /   \        /   \
%e A346979         4337  4339         4391  4397   4793  4799
%e A346979              /  |  \        |     |     /  \
%e A346979         43391 43397 43399 43913 43973 47933 47939
%e A346979                             |
%e A346979                          439133
%e A346979                             |
%e A346979                         4391339
%t A346979 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 *)
%o A346979 (Python)
%o A346979 from sympy import isprime
%o A346979 def p_count(k):
%o A346979     global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
%o A346979     for i in range(4):
%o A346979         m = 10*k + d[i]
%o A346979         if isprime(m): ct += 1; p_count(m)
%o A346979     return ct
%o A346979 for n in range(100):
%o A346979     ct = 0; print(p_count(n))
%Y A346979 Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.
%K A346979 nonn,base
%O A346979 0,1
%A A346979 _Ya-Ping Lu_, Aug 09 2021