A036378 -id:A036378 - cp's OEIS frontend

A095013 Number of 8k+-1 primes (A001132) in range [2^n,2^(n+1)].

0, 1, 0, 3, 2, 8, 10, 22, 35, 67, 126, 233, 438, 793, 1525, 2825, 5391, 10192, 19332, 36739, 70163, 133983, 256877, 492962, 946938, 1822776, 3513544, 6780795, 13102754, 25349101, 49090527, 95168113, 184659769, 358635803, 697092152, 1356042601, 2639892053, 5142809798

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A001132, A036378, A095009, A095012, A095014.

a(n) = A036378(n) - A095014(n) = A095009(n) + A095012(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A095014 Number of 8k+-3 primes (A003629) in range [2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 13, 21, 40, 70, 129, 231, 434, 819, 1505, 2884, 5358, 10198, 19303, 36847, 70173, 134233, 256831, 492856, 947182, 1822968, 3513746, 6781112, 13104524, 25348436, 49092129, 95167472, 184663536, 358631365, 697100084, 1356061232, 2639871771, 5142831980

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A003629, A036378, A095010, A095011, A095013.

a(n) = A036378(n) - A095013(n) = A095010(n) + A095011(n).

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A095334 Number of A095314-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 0, 3, 3, 5, 4, 21, 32, 69, 96, 229, 335, 768, 1116, 2860, 4371, 10252, 15490, 36563, 58041, 133739, 209875, 491193, 795599, 1816561, 2951789, 6772098, 11144763, 25284670, 41781268, 94895078, 158643268

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 0, 0.5, 0, 0.6, 0.428571, 0.384615, 0.173913, 0.488372, 0.426667, 0.50365, 0.376471, 0.493534, 0.384174, 0.476427, 0.368317, 0.500963, 0.406642, 0.502795, 0.400932, 0.496874, 0.413586, 0.498624, 0.408549, 0.498259, 0.420036, 0.498269, 0.420047, 0.499347, 0.425255, 0.498736, 0.425546, 0.498567, 0.429551

Ratios a(n)/A095297(n) converge as: 1, 1, 1, 1.5, 1.5, 0.625, 0.571429, 0.954545, 1.185185, 1.014706, 1.2, 0.974468, 0.976676, 0.909953, 0.945763, 1.003861, 0.977197, 1.011245, 1.006694, 0.987575, 0.988538, 0.994512, 0.983496, 0.993061, 0.991634, 0.9931, 0.995506, 0.997392, 0.996345, 0.994955, 0.993649, 0.994285, 0.995042

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095335(n). Cf. A095298.

A095742 Sum of A037888(p) for all primes p such that 2^n < p < 2^(n+1).

Original entry on oeis.org

0, 0, 2, 3, 9, 16, 35, 69, 148, 271, 628, 1167, 2629, 4830, 10597, 20083, 42928, 81579, 174223, 331314, 701382, 1340756, 2825575, 5422454, 11361615, 21873923, 45673361, 88161666, 183458213, 354899159, 736343490, 1427495050, 2954560104

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) gives the average asymmetricity ratio for n-bit primes: 0, 0, 1, 0.6, 1.285714, 1.230769, 1.521739, 1.604651, 1.973333, 1.978102, 2.462745, 2.515086, 3.014908, 2.996278, 3.49736, 3.517779, 3.993674, 4.000932, 4.50946, 4.502405, 4.997877, 4.998792, 5.500352, 5.500462, 5.998361, 5.999852, 6.499427, 6.500684, 7.000277, 7.000323, 7.499731, 7.499885, 7.999929, etc. I.e. 2- and 3-bit odd primes are all palindromes, 4-bit primes need on average just a one-bit flip to become palindromes, etc.

Ratio (a(n)/A036378(n))/f(n), where f(n) is (n-1)/4 if n is odd and (n-2)/4 if n is even (i.e. it gives the expected asymmetricity for all odd numbers in range [2^n,2^(n+1)]) converges as follows: 1, 1, 2, 1.2, 1.285714, 1.230769, 1.014493, 1.069767, 0.986667, 0.989051, 0.985098, 1.006034, 1.004969, 0.998759, 0.999246, 1.00508, 0.998418, 1.000233, 1.002102, 1.000535, 0.999575, 0.999758, 1.000064, 1.000084, 0.999727, 0.999975, 0.999912, 1.000105, 1.00004, 1.000046, 0.999964, 0.999985, 0.999991, ...

			a(1)=0, as only prime in range ]2,4] is 3, which has palindromic binary expansion 11, i.e. A037888(3)=0. a(2)=0, as in range ]4,8] there are two primes 5 (101 in binary) and 7 (111 in binary) so A037888(5) + A037888(7) = 0. a(3)=2, as in range ]8,16] there are two primes, 11 (1011 in binary) and 13 (1101 in binary), thus A037888(11) + A037888(13) = 1+1 = 2.

A. Karttunen, J. Moyer: C-program for computing the initial terms of this sequence

Cf. A095298, A095732 (sums of similar asymmetricity measures for Zeckendorf-expansion), A095753.

A095753 Number of almost base-2 palindromic primes (A095743) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 2, 3, 5, 4, 15, 18, 32, 33, 63, 81, 119, 144, 256, 318, 527, 640, 1029, 1281, 2236, 2566, 4273, 5410, 8261, 10610, 16868, 21084, 33943, 43104, 68218, 88493, 136343

Offset: 1

Antti Karttunen, Jun 12 2004

Ratio a(n)/A036378(n) converges as follows: 0, 0, 1, 0.6, 0.714286, 0.307692, 0.652174, 0.418605, 0.426667, 0.240876, 0.247059, 0.174569, 0.136468, 0.08933, 0.084488, 0.055702, 0.049028, 0.031388, 0.026634, 0.017408, 0.015933, 0.009567, 0.008318, 0.005488, 0.004361, 0.00291, 0.0024, 0.001555, 0.001295, 0.00085, 0.000695, 0.000465, 0.000369

Ratio a(n)/A095758(n) converges as follows: 1, 1, 0, 1.5, 1, 1, 3.75, 1.2, 2, 1.375, 1.909091, 1.446429, 1.652778, 1.515789, 1.718121, 1.452055, 1.636646, 1.191806, 1.570992, 1.283567, 1.708174, 1.380312, 1.534842, 1.392177, 1.547004, 1.311334, 1.573801, 1.302205, 1.521016, 1.419202, 1.570938, 1.389237, 1.546084

The second diagonal of triangle A095759. Cf. A095742.

A095757 Number of A095747-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 4, 7, 7, 10, 9, 26, 20, 43, 27, 74, 41, 112, 93, 181, 167, 495, 274, 796, 558, 1232, 935, 2602, 1512, 5164, 3275, 8689, 6309

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

Ratio a(n)/A036378(n) converges as follows: 1, 1, 1, 0.6, 0.285714, 0.461538, 0.173913, 0.162791, 0.093333, 0.072993, 0.035294, 0.056034, 0.022936, 0.026675, 0.008911, 0.012962, 0.003814, 0.005493, 0.002407, 0.00246, 0.00119, 0.001846, 0.000533, 0.000807, 0.000295, 0.000338, 0.000133, 0.000192, 0.000058, 0.000102, 0.000033, 0.000046, 0.000017

The last nonzero terms from each row of triangle A095759. Bisection: A095760.

A095766 Number of primes whose binary expansion begins '11' (A080166) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 11, 21, 37, 67, 125, 227, 431, 787, 1491, 2812, 5296, 10055, 19079, 36343, 69398, 132661, 254122, 488028, 937994, 1806147, 3482463, 6722625, 12994889, 25145151, 48709705, 94451647, 183312229, 356089665, 692285717

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

I.e. number of primes p such that (2^n + 2^(n-1)) < p < 2^(n+1).

Ratio a(n)/A036378(n) converges as follows: 1, 0.5, 0.5, 0.4, 0.428571, 0.538462, 0.478261, 0.488372, 0.493333, 0.489051, 0.490196, 0.489224, 0.494266, 0.488213, 0.492079, 0.492556, 0.492697, 0.493134, 0.493827, 0.493885, 0.494513, 0.494605, 0.494682, 0.495049, 0.495214, 0.495412, 0.495563, 0.495699, 0.49585, 0.495984, 0.496113, 0.496237, 0.496346

a(n) = A036378(n)-A095765(n).

f[n_] := PrimePi[2^(n + 1)] - PrimePi[2^n + 2^(n - 1) - 1]; Array[f, 35] (* Robert G. Wilson v *)

a(n)=primepi(2^(n+1))-primepi(2^n+2^(n-1)-1) \\ Charles R Greathouse IV, Sep 25 2012

a(34) and a(35) from Robert G. Wilson v, Jan 24 2006

A190502 Number of Ramanujan primes <= 2^n.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 7, 13, 23, 42, 75, 137, 255, 463, 872, 1612, 3030, 5706, 10749, 20387, 38635, 73584, 140336, 268216, 513705, 985818, 1894120, 3645744, 7027290, 13561906, 26207278, 50697533, 98182656, 190335585, 369323301, 717267167, 1394192236, 2712103833

Offset: 0

John W. Nicholson, May 11 2011

nonn

Dana Jacobsen, Table of n, a(n) for n = 0..56

Cf. A007053, A036378, A104272, A181671, A190501.

\\ With RR[.] is a list of A104272(.). The output of this program is n, a(n), and RR[a(n)].
j=0; while(2^jJohn W. Nicholson, Dec 01 2012

use ntheory ":all"; sub a190502 { scalar(@{ramanujan_primes(1 << shift)}) } say a190502($) for 0..20; # _Dana Jacobsen, Dec 19 2015

use ntheory ":all"; my $t = 0; for my $e (1..32) { $t += scalar(@{ramanujan_primes(2**($e-1)+1,2**$e)}); say "$e $t" } # Dana Jacobsen, Dec 19 2015

use ntheory ":all"; say ramanujan_prime_count(2**$) for 0..47; # _Dana Jacobsen, Jan 03 2016

Extended by T. D. Noe, May 11 2011
Extended to n = 32 by John W. Nicholson, Dec 01 2012
a(33)-a(41) from Dana Jacobsen, Dec 19 2015

A293696 Irregular triangle read by rows in which row n lists the partial sums of prime numbers between 2^n+1 and 2^(n+1).

Original entry on oeis.org

2, 3, 5, 12, 11, 24, 17, 36, 59, 88, 119, 37, 78, 121, 168, 221, 280, 341, 67, 138, 211, 290, 373, 462, 559, 660, 763, 870, 979, 1092, 1219, 131, 268, 407, 556, 707, 864, 1027, 1194, 1367, 1546, 1727, 1918, 2111, 2308, 2507, 2718, 2941, 3168, 3397, 3630, 3869, 4110, 4361

Offset: 0

Olivier Gérard, Oct 15 2017

			The triangle begins
.
2
3
5,12
11,24
17,36,59,88,119
37,78,121,168,221,280,341
67,138,211,290,373,462,559,660,763,870,979,1092,1219

Olivier Gérard, Table of n, a(n) for n = 0..6541

Cf. A000040, A036378 (length of rows), A293697 (sequence of last item of each row).

Flatten@Table[
  Rest@FoldList[Plus, 0,
    Table[Prime[i], {i, PrimePi[2^(n)] + 1, PrimePi[2^(n + 1)]}]], {n,
    0, 8}]
Table[Accumulate[Prime[Range[PrimePi[2^n]+1,PrimePi[2^(n+1)]]]],{n,0,10}] // Flatten (* Harvey P. Dale, Dec 26 2020 *)

A309359 Median of the primes p with 2^(n-1) < p < 2^n.

Original entry on oeis.org

6, 12, 23, 47, 97, 191, 383, 761, 1523, 3049, 6107, 12252, 24376, 48877, 97777, 195659, 391623, 783257, 1566386, 3133974, 6269116, 12538053, 25082361, 50170976, 100353498, 200730129, 401498897, 803081460, 1606292647, 3212862108

Offset: 3

Hugo Pfoertner, Jul 25 2019

nonn

For n >= 3, median of the primes with n binary digits. The median of an even number of values is assumed to be defined as the arithmetic mean of the two central elements in their sorted list. The special case of the primes with two binary digits {2, 3} is excluded, because their median would be 5/2.

			a(3) = 6: 2^2 < {5, 7} < 2^3, (5 + 7)/2 = 6.
a(4) = 12: 2^3 < {11, 13} < 2^4, (11 + 13)/2 = 12
a(5) = 23: 2^4 < {17, 19, 23, 29, 31} < 2^5, median = 23.

Hugo Pfoertner, Table of n, a(n) for n = 3..63

Cf. A007053, A036378, A309329.

cp's OEIS Frontend

A095013 Number of 8k+-1 primes (A001132) in range [2^n,2^(n+1)].

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A095014 Number of 8k+-3 primes (A003629) in range [2^n,2^(n+1)].

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A095334 Number of A095314-primes in range ]2^n,2^(n+1)].

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A095742 Sum of A037888(p) for all primes p such that 2^n < p < 2^(n+1).

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A095753 Number of almost base-2 palindromic primes (A095743) in range ]2^n,2^(n+1)].

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A095757 Number of A095747-primes in range ]2^n,2^(n+1)].

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A095766 Number of primes whose binary expansion begins '11' (A080166) in range ]2^n,2^(n+1)].

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Mathematica

PARI

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A190502 Number of Ramanujan primes <= 2^n.

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Perl

Perl

Perl

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A293696 Irregular triangle read by rows in which row n lists the partial sums of prime numbers between 2^n+1 and 2^(n+1).

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Mathematica

A309359 Median of the primes p with 2^(n-1) < p < 2^n.

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