cp's OEIS Frontend

Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002

(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011

Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014

Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018

Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022

The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024

Apart from the initial 0, a subsequence of A048103. Proof: For all primes p, when i >= A000720(p), neither p itself nor p^p divides a(i) [implied by Henry Bottomley's Sep 27 2006 formula], but neither does p^p divide a(i) when 0 < i < A000720(p), as then p^p > a(i). See A074107, which gives an upper bound for this sequence. - Antti Karttunen, Nov 19 2024

Does the ratio of adjacent terms converge?

It appears that lim_{n->infinity} a(n+1)/a(n) = 1.7532... - Jon E. Schoenfield, Feb 21 2019

For n > 1, a(n) is smallest prime p = prime(k) such that no fewer than (n-1)/n of any p# consecutive integers are divisible by a prime not greater than p. Cf. A053144(k)/A002110(k). - Peter Munn, Apr 29 2017

Also, the smallest prime p such that the sum of the reciprocals of the p-smooth numbers converges to at least n. - Keith F. Lynch, Apr 29 2023

Also, if m is a random integer much larger than the square of a(n), and m is not divisible by any prime less than or equal to a(n), the probability that m is prime is n/log(m). - Keith F. Lynch, Dec 17 2023

By rewriting the sequence of sums as 1 - Product_{n>=1} (1 - 1/prime(n)), one can show that the product goes to zero and the sequence of sums converges to 1. This is interesting because the terms approach 1/(2*prime(n)) for large n, and a sum of such terms might be expected to diverge, since Sum_{n>=1} 1/(2*prime(n)) diverges.

Denominators appear to be given by A060753(n+1). - Peter Kagey, Jun 08 2019

A254196 appears to be a duplicate of this sequence. - Michel Marcus, Aug 05 2019

Square array read by descending antidiagonals; A(n,k) with rows n >= 0, columns k >= 0. Prime factors are counted with multiplicity. Primorial(n) = A002110(n): product of first n primes.

Visualize the prime factorizations of the positive integers as a table with row headings giving each successive integer, and the primes of which the row heading is the product listed across the columns in nondecreasing order, repeated when necessary. Except for 1, which lacks prime factors, column 1 has the row heading's least prime factor, column 2 has a value for composite numbers but is blank for primes, and so on. This sequence measures precisely how frequently values up to and including the various primes occur in each column. This is possible because any given prime occurs cyclically in any given column, for the reason following.

The occurrence pattern of up to k factors of prime(n) in such prime factorizations has a fundamental period over the positive integers of prime(n)^k. The least common period for up to k factors of each of the first n primes is primorial(n)^k, and this covers everything that can affect the occurrence of prime(n) in the least k factors. Thus prime(n) is k-th least prime factor of integer m if and only if it is k-th least prime factor of m + primorial(n)^k.

Intermediate values in the calculation of this sequence appear in A281890.

If n > 0, A(n,1) = A053144(n) in accordance with the comment on A053144 dated Apr 08 2010.

A(2,k) = A027649(k) = 2*(3^k) - 2^k.

Here prime(n)# denotes the product of the first n primes. Row n begins with A005867(n-1). The other n-1 terms in row n are prime(n) times the previous row. The sum of the terms in row n is cototient(prime(n)#), which is A053144(n), and which equals prime(n)#-A005867(n). This sequence is a generalization of a comment in A005867 by Dennis Martin.

T(n,k) is the triangle in A174909 with reversed row order. (See that sequence for additional comments).

Row sums = A053144(n) = A002110(n) - T(n+1,n+1).

T(n,k) = number of terms with smallest prime factor prime(k) contained in primorial(n) consecutive numbers, k <= n. For example, T(5,4) = 88, so there are 88 terms with smallest prime factor 7 in any sequence of 2310 consecutive numbers.