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A036378
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Number of primes p between powers of 2, 2^n < p <= 2^(n+1).
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87
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1, 1, 2, 2, 5, 7, 13, 23, 43, 75, 137, 255, 464, 872, 1612, 3030, 5709, 10749, 20390, 38635, 73586, 140336, 268216, 513708, 985818, 1894120, 3645744, 7027290, 13561907, 26207278, 50697537, 98182656, 190335585, 369323305, 717267168, 1394192236
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of primes whose binary order (A029837) is n, i.e. those with ceiling[ Log[ 2,p ] ] = n.
First differences of A007053. This sequence illustrates how far the Bertrand postulate is over satisfied.
Scaled for Ramanujan primes as in A190501, A190502.
This sequence appears complete such that any nonnegative number can be written as a sum of distinct terms of this sequence. The sequence has been checked for completeness up to the gap between 2^46 and 2^47. Assuming that after 2^46 the formula x/log(x) is a good approximation to primepi(x), it can be proved that 2*a(n) > a(n+1) for all n >= 46, which is a sufficient condition for completeness. [Frank M Jackson, Feb 02 2012]
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REFERENCES
| Seung-Hoon Lee, Mario Gerla, Hugo Krawczyk, Kang-Won Lee, and Elizabeth A. Quaglia, Performance Evaluation of Secure Network Coding using Homomorphic Signature, http://www.cs.ucla.edu/~shlee/papers/netcod_TECH.pdf
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..74 (using data from A007053)
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]
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FORMULA
| a(n)=primepi(2^n)-primepi(2^(n-1))
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EXAMPLE
| The 7 primes for which A029837(p)=6 are 37,41,43,47,53,59,61.
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MATHEMATICA
| t = Table[PrimePi[2^n], {n, 0, 20}]; Rest@t - Most@t (* Robert G. Wilson v *)
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PROG
| (Pari) a(n) = primepi(1<<n)-primepi(1<<(n-1))
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CROSSREFS
| a(n) = A095005(n)+A095006(n) = A095007(n) + A095008(n) = A095013(n) + A095014(n) = A095015(n) + A095016(n) (for n > 1) = A095021(n)+A095022(n)+A095023(n)+A095024(n) = A095019(n)+A095054(n)
= A095020(n)+A095055(n) = A095060(n)+A095061(n) = A095063(n)+A095064(n) = A095094(n)+A095095(n).
Cf. A000720, A190501, A190502, A190568, A007053
Sequence in context: A095333 A095326 A095330 * A028303 A195964 A047083
Adjacent sequences: A036375 A036376 A036377 * A036379 A036380 A036381
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu)
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EXTENSIONS
| More terms from Labos E. (labos(AT)ana.sote.hu), May 13 2004
Entries checked by Robert G. Wilson v (rgwv(at)rgwv.com), Mar 20 2006
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