cp's OEIS Frontend

a(n) = signum(A066246(n)), where signum = A057427. For n > 1: a(n) = 1 - A010051(n) = A005171(n).

a(n) = A057427(A086971(n)). - Reinhard Zumkeller, Dec 14 2012

a(n) < log(4)*n/log(n) < 7*n/(5*log(n)) for n > 1. - Reinhard Zumkeller, Mar 04 2008

Bertrand's postulate is equivalent to the formula a(n) >= 1 for all positive integers n. - Jonathan Vos Post, Jul 30 2008

Number of distinct prime factors > n of binomial(2*n,n). - T. D. Noe, Aug 18 2011

f(2, 2n) - f(3, n) < a(n) < f(3, 2n) - f(2, n) for n > 5889 where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). The constant 3 can be improved. - Charles R Greathouse IV, May 02 2012

For n >= 2, a(n) is the number of primes appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.

Also, least k such that 2n = A001223(k-1) = prime(k+1) - prime(k), where prime(k) = A001223(n). - Alexander Adamchuk, Jul 30 2006

Also the least number k>0 such that the k-th maximal run of composite numbers has length 2n-1. For example, the 8th such run (24,25,26,27,28) is the first of length 2(3)-1, so a(3) = 8. Also positions of first appearances in A176246 (A046933 without first term). - Gus Wiseman, Jun 12 2024

We consider 1 to be a power of a prime and a non-prime-power, but not a prime-power.

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.

S reads like this:

"At position 2, there is a prime in S" [indeed, this is 3]

"At position 3, there is a prime in S" [indeed, this is 5]

"At position 5, there is a prime in S" [indeed, this is 7]

"At position 1, there is a prime in S" [indeed, this is 2]

"At position 7, there is a prime in S" [indeed, this is 11]

"At position 8, there is a prime in S" [indeed, this is 13]

"At position 11, there is a prime in S" [indeed, this is 19]

"At position 13, there is a prime in S" [indeed, this is 23]

"At position 10, there is a prime in S" [indeed, this is 17], etc.

S is built with this rule: when you are about to write a term of S, always use the smallest integer not yet present in S and not leading to a contradiction.

Thus one cannot start with 1; this would read: "At position 1, there is a prime number in S" [no, 1 is not a prime]

So start S with 2 and the rest follows smoothly.

S contains all the primes and they appear in their natural order.

Does the ratio primes/composites in S tend to a limit?

The definition and the comments above are Eric Angelini's original submission. A more formal definition would be "Lexicographically earliest sequence of distinct positive numbers such that k is a term of the sequence iff a(k) is a prime". However, to honor Eric Angelini's memory, we will retain his enigmatic definition. - N. J. A. Sloane, Dec 20 2024

Comments from N. J. A. Sloane, Nov 14 2024 (Start)

Theorem. Let p(k) = k-th prime, c(k) = k-th composite number. For n >= 5, if n is a prime or n = c(2*t+1) for some t, then a(n) = p(k) where k = floor((n+PrimePi(n))/2); otherwise, n = c(2*t) for some t and a(n) = c(2*t+1).

The proof will be added later (see reference).

The theorem implies that the sequence consists of the primes and the odd-subscripted composite numbers.

All of Dean Hickerson's comments below follow from this theorem. (End)

Comments from Dean Hickerson, Aug 11 2006: (Start)

In the limit, exactly half of the terms are primes. Here's a formula, found empirically, for a(n) for n >= 5:

Let pi(n) be the number of primes <= n and p(n) be the n-th prime. Then for n >= 5:

- if n is prime or (n is composite and n+pi(n) is even) then a(n) = p(floor((n+pi(n))/2));

- if n is composite and n+pi(n) is odd and n+1 is composite then a(n) = n+1;

- if n is composite and n+pi(n) is odd and n+1 is prime then a(n) = n+2.

Also, for n >= 5, a(n) is in the sequence iff either n is prime or n+pi(n) is even.

(This could all be proved by induction on n.)

It follows from this that, for n >= 4, the number of primes among a(1), ..., a(n) is exactly floor((n+pi(n))/2). Since pi(n)/n -> 0 as n -> infinity, this is asymptotic to n/2. (End)

a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - Sean A. Irvine, Apr 18 2023

Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - Antti Karttunen, Dec 08 2024

All the permutations A091202, A091204, A106442, A106444, A106446, A235041 share the same property that primes (A000040) are mapped bijectively to the binary representations of irreducible GF(2) polynomials (A014580) but while they determine the mapping of composites (A002808) to the corresponding binary codes of reducible polynomials (A091242) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

The number of composite numbers <= n is n less the number of primes less 1.

The sequence is nondecreasing.