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A095765
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Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).
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3
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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
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internal format)
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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
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MATHEMATICA
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a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited, restoring meaning of name, by Peter Munn, Jun 27 2023
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STATUS
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approved
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