A095062 -id:A095062 - cp's OEIS frontend

A095321 Primes in whose binary expansion the number of 1-bits is <= number of 0-bits minus 3.

257, 521, 577, 641, 769, 1031, 1033, 1049, 1061, 1091, 1093, 1097, 1153, 1217, 1283, 1289, 1297, 1409, 1553, 1601, 2053, 2069, 2081, 2083, 2089, 2113, 2129, 2179, 2309, 2593, 2689, 3089, 3137, 3329, 4099, 4111, 4129, 4133, 4139, 4153

Offset: 1

Antti Karttunen, Jun 04 2004

Robert Israel, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Complement of A095320 in A000040. Subset of A095317. Cf. also A095331.

filter:= proc(p) local L;
  L:= convert(p,base,2);
  2*convert(L,`+`) <= nops(L)-3
end proc;
select(filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Apr 27 2025

Select[Prime[Range[600]],DigitCount[#,2,1]<=DigitCount[#,2,0]-3&] (* Harvey P. Dale, Jul 04 2018 *)

forprime(p=2, 4200, v=binary(p); s=0; for(k=1,#v, s+=if(v[k]==1,+1,-1)); if(s<=-3, print1(p,", "))) \\ Washington Bomfim, Jan 12 2011

A095322 Primes in whose binary expansion the number of 1 bits is > 4 + number of 0 bits.

Original entry on oeis.org

31, 127, 191, 223, 239, 251, 367, 379, 383, 431, 439, 443, 463, 479, 487, 491, 499, 503, 509, 751, 863, 887, 983, 991, 1013, 1019, 1021, 1151, 1277, 1279, 1399, 1439, 1471, 1487, 1499, 1511, 1523, 1531, 1663, 1723, 1759, 1783, 1787, 1789, 1823

Offset: 1

Antti Karttunen, Jun 04 2004

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

Complement of A095323 in A000040. Subset of A095318. Subset: A095284. Cf. also A095324.

btsQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Count[idn2,1]>4+Count[ idn2,0]]; Select[Prime[Range[300]],btsQ] (* Harvey P. Dale, Nov 12 2011 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 > (4+b0), return(1);, return(0););};
forprime(x = 2, 1823, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 12 2011

A095323 Primes in whose binary expansion the number of 1 bits is <= 4 + number of 0 bits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 227, 229, 233, 241, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Antti Karttunen, Jun 04 2004

Differs from primes (A000040) first time at n=11, where a(11)=37, while A000040(11)=31, as 31 whose binary expansion is 11111, with five 1 bits and no 0 bits is the first prime excluded from this sequence. Note that 15 (1111 in binary) is not prime.

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

Complement of A095322 in A000040. Subset of A095285. subset: A095319. Cf. A095325.

Select[Prime[Range[100]],DigitCount[#,2,1]<(5+DigitCount[#,2,0])&] (* Harvey P. Dale, Dec 09 2015 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b1 <= (4+b0), return(1);, return(0););};
forprime(x = 2, 293, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 12 2011

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.

A095743 Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.

Original entry on oeis.org

2, 11, 13, 19, 23, 29, 37, 41, 47, 59, 61, 67, 89, 97, 103, 131, 137, 157, 167, 173, 181, 191, 193, 199, 211, 223, 227, 229, 239, 251, 277, 281, 317, 337, 349, 367, 373, 383, 401, 419, 431, 463, 467, 479, 487, 491, 503, 509, 521, 563, 569, 577

Offset: 1

Antti Karttunen, Jun 12 2004

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

The second row of array A095749. Cf. A095753, A095748.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(abs(L[i]-L[-i]),i=1..floor(nops(L)/2))
end proc:
select(t -> f(t) = 1, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Dec 04 2023

A095748 Almost maximally asymmetric primes in binary.

Original entry on oeis.org

17, 31, 37, 41, 47, 59, 61, 67, 89, 97, 103, 139, 149, 163, 197, 263, 269, 283, 293, 307, 353, 359, 379, 389, 409, 433, 439, 449, 461, 499, 541, 557, 607, 613, 631, 659, 727, 743, 809, 829, 877, 929, 941, 953, 997, 1009, 1039, 1051, 1151, 1171

Offset: 1

Antti Karttunen, Jun 12 2004

Primes p for which A037888(p) = floor((A070939(p)-4)/2). Those numbers contain just two bits mirroring each other, beyond the first and last bits. (All the odd primes without leading zeros begin and end in 1 bits.)

			a(5)=(101111)2. In this case, from left to right, the third bit agrees with the fourth. The prime 53 = (110101)_2 is not a term since the symmetry is limited to the first and last bits.

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

Cf. A095758, A095749, A095743.

A070939(p) = { return(floor(log(p)/log(2))+1) };
A037888(p)={v=binary(p);s=0;j=#v;for(k=1,#v,s+=abs(v[k]- v[j]);j--);return(s/2);}; forprime(p=3,1171,if(A037888(p)==floor((A070939(p)-4)/2), print1(p,", ")))

Edited by Washington Bomfim, Jan 13 2011

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

cp's OEIS Frontend

A095321 Primes in whose binary expansion the number of 1-bits is <= number of 0-bits minus 3.

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A095322 Primes in whose binary expansion the number of 1 bits is > 4 + number of 0 bits.

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A095323 Primes in whose binary expansion the number of 1 bits is <= 4 + number of 0 bits.

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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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A095743 Primes p for which A037888(p) = 1, i.e., primes whose binary expansion is almost symmetric, needing just a one-bit flip to become palindrome.

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Maple

A095748 Almost maximally asymmetric primes in binary.

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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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