A087724 -id:A087724 - cp's OEIS frontend

A087723 Values that appear more than once in A087724 (related to the growth of PrimePi).

1, 2, 7, 11, 12, 13, 15, 19, 26, 29, 33, 35, 41, 48, 54, 71, 72, 76, 84, 86, 90, 92, 93, 96, 98, 103, 105, 108, 117, 122, 127, 129, 130, 137, 142, 152, 157, 158, 165, 176, 180, 183, 191, 193, 194, 196, 213, 214, 218, 219, 229, 231, 233, 234, 235, 246, 250, 256, 267

Offset: 1

Roger L. Bagula, Sep 29 2003

nonn

Digits=1000 a[n_]=-PrimePi[n]+Floor[Prime[n]/Log[n]]-2 b=Table[a[n], {n, 2, Digits}] c=Table[If[b[[n]]-b[[n-1]]==0, b[[n]], 0], {n, 2, Digits-1}] d=Delete[Union[c], 1]

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.

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A087723 Values that appear more than once in A087724 (related to the growth of PrimePi).

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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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