login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165). 3
0, 1, 1, 3, 4, 6, 12, 22, 38, 70, 130, 237, 441, 825, 1539, 2897, 5453, 10335, 19556, 37243, 70938, 135555, 259586, 497790, 956126, 1839597, 3544827, 6839282, 13212389, 25552386, 49472951, 95883938, 186011076, 361177503, 701906519 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
I.e., number of primes p such that 2^n < p < (2^n + 2^(n-1)).
Ratio a(n)/A036378(n) converges as follows: 0, 0.5, 0.5, 0.6, 0.571429, 0.461538, 0.521739, 0.511628, 0.506667, 0.510949, 0.509804, 0.510776, 0.505734, 0.511787, 0.507921, 0.507444, 0.507303, 0.506866, 0.506173, 0.506115, 0.505487, 0.505395, 0.505318, 0.504951, 0.504786, 0.504588, 0.504437, 0.504301, 0.50415, 0.504016, 0.503887, 0.503763, 0.503654
Ratio a(n)/A095766(n) converges as follows: 0, 1, 1, 1.5, 1.333333, 0.857143, 1.090909, 1.047619, 1.027027, 1.044776, 1.04, 1.044053, 1.023202, 1.048285, 1.032193, 1.030228, 1.029645, 1.027847, 1.025001, 1.024764, 1.022191, 1.021815, 1.021501, 1.020003, 1.019331, 1.01852, 1.017908, 1.017353, 1.016737, 1.016195, 1.015669, 1.015164, 1.014723
I think this explains also the bias present in ratios shown at A095297, A095298, etc.
LINKS
FORMULA
a(n) = A036378(n)-A095766(n).
EXAMPLE
Table showing the derivation of the initial terms:
n 2^n+1 2^(n+1) a(n) primes starting '10' in binary
1 3 4 0 -
2 5 8 1 5 = 101_2
3 9 16 1 11 = 1011_2
4 17 32 3 17 = 10001_2, 19 = 10011_2, 23 = 10111_2
MATHEMATICA
a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
Array[a, 35] (* Robert G. Wilson v, Jan 24 2006 *)
CROSSREFS
Sequence in context: A329671 A124571 A332279 * A095016 A358358 A160684
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 12 2004
EXTENSIONS
a(34) and a(35) from Robert G. Wilson v, Jan 24 2006
Edited, restoring meaning of name, by Peter Munn, Jun 27 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)