A085581 -id:A085581 - cp's OEIS frontend

A087724 a(n) = -PrimePi(n) +floor( prime(n)/log(n))-2.

1, 0, 1, 1, 2, 2, 3, 4, 6, 5, 7, 7, 8, 9, 11, 11, 12, 12, 13, 13, 15, 15, 17, 19, 19, 20, 21, 20, 21, 23, 24, 26, 26, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 35, 35, 37, 40, 41, 41, 42, 43, 42, 44, 46, 47, 48, 48, 48, 49, 48, 50, 54, 54, 54, 55, 57, 58, 60, 61, 60, 61, 62, 63, 64

Offset: 2

Roger L. Bagula, Sep 29 2003

nonn

A087724 := proc(n)
    floor( ithprime(n)/log(n))-numtheory[pi](n) -2 ;
end proc: # R. J. Mathar, May 15 2013

Digits=200 a[n_]=-PrimePi[n]+Floor[Prime[n]/Log[n]]-2 b=Table[a[n], {n, 2, Digits}]

a(n) = A085581(n)-A000720(n) -2. - R. J. Mathar, May 15 2013

A250621 a(n) = floor(n*log(prime(n))).

Original entry on oeis.org

0, 2, 4, 7, 11, 15, 19, 23, 28, 33, 37, 43, 48, 52, 57, 63, 69, 73, 79, 85, 90, 96, 101, 107, 114, 119, 125, 130, 136, 141, 150, 156, 162, 167, 175, 180, 187, 193, 199, 206, 212, 218, 225, 231, 237, 243, 251, 259, 265, 271, 278, 284, 290, 298, 305, 312, 318, 324, 331

Offset: 1

Freimut Marschner, Nov 26 2014

From n < prime(n), n >= 1 follows that n*log(n) < prime(n) < n*log(prime(n)), n >= 4. This inequality is included in the prime number theorem PNT.

			For n = 1, prime(1) = 2, floor(1*0.69... = 0.69...) = 0 ;
For n = 25, prime(25) = 97, floor(25*4.57... = 114.36...) = 114.

Cf. A050504 (floor(n*log(n))), A086861 (floor(prime(n)/log(prime(n)))), A085581 (floor(prime(n)/log(n))), A050504 (integer part of n*log(n)), A050503 (nearest integer to n*log(n)), A050502 (ceiling of n*log(n)).

Table[Floor[n Log[Prime[n]]],{n,60}] (* Harvey P. Dale, Aug 13 2019 *)

vector(100,n,floor(n*log(prime(n)))) \\ Derek Orr, Nov 28 2014

A251482 a(n) = floor(prime(n)/log(n)) + ceiling(prime(n)/log(prime(n))) - 2*n, n >=2.

Original entry on oeis.org

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

Offset: 2

Freimut Marschner, Dec 07 2014

The prime number theorem implies prime(n)/log(prime(n)) < n < prime(n)/log(n), n >= 2. From this follows a(n).

			a(4) = floor(5.04...) + ceiling(3.59...) - 2*4 = 5 + 4 - 2*4 = 1.

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A086861 (floor(prime(n)/log(prime(n)))), A085581 (floor(prime(n)/log(n))).
Cf. A087724 (-PrimePi(n) + floor(prime(n)/log(n)) - 2), A000720 (pi(n)).
Cf. A060715 (Number of primes between n and 2n exclusive).

[Floor(NthPrime(n)/Log(n)) + Ceiling(NthPrime(n)/Log(NthPrime(n))) - 2*n: n in [2..100]]; // Vincenzo Librandi, Mar 25 2015

a251482[n_Integer] :=
Floor[Prime[#]/Log[#]] + Ceiling[Prime[#]/Log[Prime[#]]] - 2 # & /@
Range[2, n]; a251482[100] (* Michael De Vlieger, Dec 15 2014 *)

vector(100,n,floor(prime(n+1)/log(n+1))+ceil(prime(n+1)/log(prime(n+1)))-2*n-2) \\ Derek Orr, Dec 30 2014

a(n) = A085581(n) + (A086861(n) + 1) - 2*n.

A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

Original entry on oeis.org

1, -1, -4, 6, 1, 8, 1, 8, 15, 22, 11, 19, 6, 14, 22, 31, 14, 23, 4, 14, 24, 34, 11, 22, 33, 44, 55, 67, 38, 50, 19, 31, 44, 57, 70, 83, 46, 60, 74, 88, 47, 62, 19, 34, 50, 66, 19, 35, 51, 68, 85, 102, 49, 67, 85, 103, 121, 139, 80, 99, 38, 57, 77, 97, 117, 137, 70

Offset: 1

Scott R. Shannon, Jan 02 2020

The Prime Number Theorem shows that the probability of a random number not greater than x being prime is approximately 1/log(x), therefore the probability of a number being composite in the same range is approximately (log(x)-1)/log(x). As this sequence subtracts n from the previous term if n is prime, or adds n with a weighting of 1/(log(n)-1) if n is composite, its expected value as n goes to infinity is approximately n*(1/(log(n)-1))*((log(n)-1)/log(n)) - n*(1/log(n)) = 0. We therefore expect that a(n)/n approaches 0 as n goes to infinity.

In the first 2 million terms the sequence changes sign 1900 times, has a maximum positive value of 160213275 at a(1772200), and a maximum negative value of -29535301 at a(1513751). The majority of terms are positive. See the image link below.

Scott R. Shannon, Graph showing a(n) for n = 1 to 2000000. The blue line is the x axis.
Wikipedia, Prime Number Theorem and Prime Counting Function.

Cf. A000040, A057835, A060769, A060770, A050499, A085581.

a[1] = 1; a[n_] := a[n] = a[n - 1] + If[PrimeQ[n], -n, Floor[n/(Log[n] - 1)]]; Array[a, 67] (* Amiram Eldar, Jan 05 2020 *)

cp's OEIS Frontend

A087724 a(n) = -PrimePi(n) +floor( prime(n)/log(n))-2.

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A250621 a(n) = floor(n*log(prime(n))).

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A251482 a(n) = floor(prime(n)/log(n)) + ceiling(prime(n)/log(prime(n))) - 2*n, n >=2.

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A330823 a(1) = 1; for n > 1, a(n) = a(n-1) - n if n is prime, otherwise a(n) = a(n-1) + floor(n/(log(n)-1)).

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