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

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, 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, 1, 1, 2, 1, 2, 0, 2, 2, 3, 4, 3, 2, 0, 6, 1, 1, 4, 4

Offset: 1

Author

A.H.M. Smeets, Dec 25 2020

Keywords

• nonn

Comments

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.

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.

Examples

			The prime number 11 occurs 2 times in A033932, and A000040(5) = 11, so a(5) = 2.
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..512
• A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Crossrefs

Cf. A000040, A033932, A033933.

Formula

It seems that Sum_{k = 1..n} a(k) ~ 0.7*A000040(n)/log(log(A000040(n))).