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A339959 Number of times the n-th prime (=A000040(n)) occurs in A033932.

%I A339959 #16 Jan 18 2021 18:09:22
%S A339959 0,0,1,2,2,0,1,1,2,2,3,2,1,2,1,3,1,2,3,3,1,1,2,1,5,2,1,4,1,3,4,6,1,2,
%T A339959 3,0,1,0,1,0,0,3,2,1,1,1,0,3,4,5,1,5,5,0,3,0,0,8,1,0,5,2,3,2,1,4,5,1,
%U A339959 1,1,2,1,2,0,2,2,3,4,3,2,0,6,1,1,4,4
%N A339959 Number of times the n-th prime (=A000040(n)) occurs in A033932.
%C A339959 Each term in A033932 is either 1 or a prime number. Moreover, it is known that each prime occurs only a finite number of times in A033932.
%C A339959 By excluding the terms that equal one from A033932, we observe the smallest value of A033933(n)/log(n!) in the range n = 2..4000 to be ~0.1399. From this it is believed that the primes less than 0.9*log(4001!)*0.1399 (~ 3676) will not occur anymore in the sequence A033932 for n > 4000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 3676 will no longer occur in A033932 for n > 4000.
%H A339959 A.H.M. Smeets, <a href="/A339959/b339959.txt">Table of n, a(n) for n = 1..512</a>
%H A339959 A.H.M. Smeets, <a href="/A339959/a339959.png">Sum_{k = 1..n} a(k) versus A000040(n)</a>
%F A339959 It seems that Sum_{k = 1..n} a(k) ~ 0.7*A000040(n)/log(log(A000040(n))).
%e A339959 The prime number 11 occurs 2 times in A033932, and A000040(5) = 11, so a(5) = 2.
%Y A339959 Cf. A000040, A033932, A033933.
%K A339959 nonn
%O A339959 1,4
%A A339959 _A.H.M. Smeets_, Dec 25 2020