A095297 -id:A095297 - cp's OEIS frontend

A095298 Sum of 1-bits between the most and least significant bits summed for all primes in range ]2^n,2^(n+1)].

0, 1, 2, 8, 15, 30, 67, 154, 302, 611, 1280, 2546, 5207, 10447, 21123, 42783, 85726, 173102, 347243, 698544, 1401784, 2813930, 5644165, 11328192, 22712057, 45538473, 91288241, 182965151, 366691833, 734702678, 1471976078, 2948741819

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratio a(n)/A036378(n) (i.e. average number of 1-bits in range ]most significant bit,least significant bit[ of primes p which 2^n < p < 2^(n+1)) grows as: 0, 0.5, 1, 1.6, 2.142857, 2.307692, 2.913043, 3.581395, 4.026667, 4.459854, 5.019608, 5.487069, 5.97133, 6.480769, 6.971287, 7.493957, 7.975254, 8.489554, 8.987783, 9.492893, 9.98877, 10.491283, 10.987107, 11.49116, 11.990823, 12.490859, 12.990533, 13.491108, 13.991985, 14.491881, 14.992221, 15.492331, 15.992713.

Ratio of that average compared to (n-1)/2 (the expected value of that same sum computed for all odd numbers in the same range) converges as: 1, 1, 1, 1.066667, 1.071429, 0.923077, 0.971014, 1.023256, 1.006667, 0.991079, 1.003922, 0.997649, 0.995222, 0.997041, 0.995898, 0.999194, 0.996907, 0.998771, 0.998643, 0.999252, 0.998877, 0.99917, 0.998828, 0.999231, 0.999235, 0.999269, 0.999272, 0.999341, 0.999427, 0.99944, 0.999481, 0.999505, 0.999545.

			a(1)=0, as only prime in range ]2,4] is 3, 11 in binary which has no space between its most and least significant bits. a(2)=1, as in that range there are two primes 5 (101 in binary) and 7 (111 in binary) and summing their middle bits we get 1. a(3)=2, as there are again two primes, 11 (1011 in binary) and 13 (1101 in binary) and summing the bits in the middle we get total 2.

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

A095297, A095334. Cf. also A095353 (similar sums and ratios computed in Fibonacci number system).

A095287 Primes in whose binary expansion the number of 1-bits is <= 1 + number of 0-bits.

Original entry on oeis.org

2, 5, 17, 19, 37, 41, 67, 71, 73, 83, 89, 97, 101, 113, 131, 137, 139, 149, 163, 193, 197, 257, 263, 269, 271, 277, 281, 283, 293, 307, 313, 331, 337, 353, 389, 397, 401, 409, 419, 421, 433, 449, 457, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 613

Offset: 1

Antti Karttunen, Jun 04 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
5 is in the sequence because 5_10 = 101_2. '101' has two 1's and one 0.
17 is in the sequence because 17_10 = 10001_2. '10001' has two 1's and three 0's. (End)

Indranil Ghosh, Table of n, a(n) for n = 1..25000
A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Complement of A095286 in A000040. Subset: A095075. Subset of A095315. Cf. also A095297.

Select[Prime[Range[200]],DigitCount[#,2,1]<=1+DigitCount[#,2,0]&] (* Harvey P. Dale, Apr 18 2023 *)

forprime(p=2,613,v=binary(p);s=0;for(k=1,#v,s+=if(v[k]==1,+1,-1));if(s<=1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

from sympy import isprime
i=1
j=1
while j<=250:
    if isprime(i) and bin(i)[2:].count("1")<=1+bin(i)[2:].count("0"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

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.

A095296 Number of A095286-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 16, 21, 48, 69, 175, 229, 529, 768, 1850, 2860, 6276, 10252, 23248, 36563, 81622, 133739, 300311, 491193, 1091809, 1816561, 4062176, 6772098, 15021634, 25284670, 56134342, 94895078, 209889612

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 0.5, 1, 0.6, 0.714286, 0.384615, 0.695652, 0.488372, 0.64, 0.50365, 0.686275, 0.493534, 0.606651, 0.476427, 0.610561, 0.500963, 0.583868, 0.502795, 0.601734, 0.496874, 0.581618, 0.498624, 0.584595, 0.498259, 0.57642, 0.498269, 0.578057, 0.499347, 0.573186, 0.498736, 0.571734, 0.498567, 0.568309

Ratios a(n)/A095335(n) converge as: 1, 1, 1, 1.5, 1.25, 0.625, 0.842105, 0.954545, 1.116279, 1.014706, 1.100629, 0.974468, 0.985102, 0.909953, 0.966562, 1.003861, 0.984008, 1.011245, 1.00445, 0.987575, 0.991822, 0.994512, 0.988408, 0.993061, 0.99389, 0.9931, 0.99673, 0.997392, 0.997286, 0.994955, 0.995265, 0.994285, 0.996248

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)-A095297(n). Cf. A095298.

A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 12 2004

nonn

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.

			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

Antti 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)]

Cf. A036378, A080165, A095297, A095298, A095766.

a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
Array[a, 35] (* Robert G. Wilson v, Jan 24 2006 *)

a(n) = A036378(n)-A095766(n).

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

Edited, restoring meaning of name, by Peter Munn, Jun 27 2023

cp's OEIS Frontend

A095298 Sum of 1-bits between the most and least significant bits summed for all primes in range ]2^n,2^(n+1)].

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A095287 Primes in whose binary expansion the number of 1-bits is <= 1 + number of 0-bits.

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

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A095296 Number of A095286-primes in range ]2^n,2^(n+1)].

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A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).

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