A340007 -id:A340007 - cp's OEIS frontend

A339274 Number of times the n-th prime (=A000040(n)) occurs in A033933.

0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 4, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 4, 0, 4, 2, 4, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 0, 3, 3, 3, 2, 2, 1, 0, 4, 1, 2, 0, 2, 1, 3, 2, 4, 2, 2, 3, 4, 0, 4, 1, 3, 2, 2, 4, 0, 5, 2, 6, 2, 3, 3, 0, 5, 2, 4, 2, 3, 3, 1, 3, 2

Offset: 1

A.H.M. Smeets, Dec 25 2020

nonn

Each term in A033933 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A033933.

By excluding the terms that equal one from A033933, we observe the smallest value of A033933(n)/log(n!) in the range n = 3..2000 to be ~0.1552. From this it is believed that the primes less than 0.9*log(2001!)*0.1552 (~ 1846) will not occur anymore in the sequence A033933 for n > 2000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 1846 will no longer occur in A033933.

			The prime number 13 occurs 2 times in A033933, and A000040(6) = 13, so a(6) = 2.

A.H.M. Smeets, Table of n, a(n) for n = 1..283
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A033932, A033933, A339959.

A340006 Number of times the n-th prime (=A000040(n)) occurs in A060270.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 3, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 2, 2, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2

Offset: 1

A.H.M. Smeets, Dec 26 2020

nonn

Each term in A060270 is either 1 or a prime number. Moreover it is known that each prime occurs only a finite number of times in A060270.

By excluding the terms that equal one from A060270, we observe the smallest value of A060270(n)/log(A002110(n)) in the range n = 2..1000 to be ~1.014. From this it is believed that the primes less than 0.9*log(A002110(1001))*1.014 (~ 7138) will not occur anymore in the sequence A060270 for n > 1000; the applied factor 0.9 is a safety factor to be more or less sure that the prime numbers up to about 7138 will no longer occur in A060270.

			The prime number 7 does not occur in A060270, and A000040(4) = 7, so a(4) = 0.
The prime number 11 occurs 1 time in A060270, and A000040(5) = 11, so a(5) = 1.

A.H.M. Smeets, Table of n, a(n) for n = 1..916
A.H.M. Smeets, Sum_{k = 1..n} a(k) versus A000040(n)

Cf. A000040, A002110, A038711, A060270, A340007.

cp's OEIS Frontend

A339274 Number of times the n-th prime (=A000040(n)) occurs in A033933.

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