A096512 -id:A096512 - cp's OEIS frontend

A096509 Number of prime-powers [including primes] in the (up and down) neighborhood of n with Ceiling[Log[n]] radius.

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

Offset: 1

Labos Elemer, Jul 12 2004

nonn

With increasing n the radius of log(n) slowly increases, while frequency of prime-powers decreases. Thus hard to estimate upper bound of terms in this sequence.

Heuristically a(n) = 0 about 1/e^2 = 13.53...% of the time. The first few instances are 1, 300, 324, 895, 896, 897, 898, 899, 1077, .... - Charles R Greathouse IV, Apr 30 2015

			n=284736: in [284723,284749] around n, 8 prime(powers) occur,radius=13, a[284736]=8.

Cf. A096510, A096511, A096512.

a[n_] := Select[Range[n - Ceiling[Log[n]], n + Ceiling[Log[n]]], PrimePowerQ] // Length; Array[a, 105] (* Jean-François Alcover, Oct 06 2016 *)

a(n)=my(t=ceil(log(n))); sum(k=n-t,n+t,!!isprimepower(k)) \\ Charles R Greathouse IV, Apr 29 2015

a(n) <= A023193(2*A004233(n)+1) + A000720(A000523(A004233(n) + n)) and so a(n) << log n/log log n (with constant at most 4 + 1/log(2) = 5.442...). Probably a(n) < 2 log n/log log n + O(log n/(log log n)^2). - Charles R Greathouse IV, Apr 29 2015

A096510 a(n) is the smallest number x such that the number of prime powers (including primes, excluding 1), in the neighborhood of x with radius ceiling(log(x)), equals n.

Original entry on oeis.org

1, 54, 2, 12, 3, 8, 4792, 75991, 284736, 6561003, 51448375, 964669618, 18320500423

Offset: 0

Labos Elemer, Jul 12 2004

With increasing n the radius of log(n) slowly increases, while frequency of prime-powers decreases.

No more terms < 369*10^8. - David Wasserman, Nov 16 2007

			a[8]=284736: because in [284723,284749] around a(8),
8 prime(powers) occur first,with radius=r=13;
a[0]=1;a[1]=54 means that in [50,58] only 53 is prime,r=4.

Cf. A096508-A096512.

Cf. A096509-A096523.

4 more terms from David Wasserman, Nov 16 2007

cp's OEIS Frontend

A096509 Number of prime-powers [including primes] in the (up and down) neighborhood of n with Ceiling[Log[n]] radius.

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