A080166 -id:A080166 - cp's OEIS frontend

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

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

A080165 Primes having initial digits "10" in binary representation.

Original entry on oeis.org

2, 5, 11, 17, 19, 23, 37, 41, 43, 47, 67, 71, 73, 79, 83, 89, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 521, 523, 541, 547, 557, 563

Offset: 1

Reinhard Zumkeller, Feb 03 2003

Also primes that terminate at 4,2,1 in the x-1 problem: Repeat, if x is even divide by 2 else subtract 1, until 4 is reached. - Cino Hilliard, Mar 27 2003

David W. Wilson remarks that it follows from standard results about primes in short intervals (see for example Harman, 1982) that there are infinitely many numbers in any base b starting with any nonzero prefix c. - N. J. A. Sloane, Sep 19 2015

			A000040(15)=47 -> '101111' therefore 47 is a term.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
G. Harman, Primes in short intervals, Math. Zeit., 180 (1982), 335-348.

Cf. A004676, A080167.
Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Column k=2 of A262365.

Select[Prime[Range[1000]], IntegerDigits[#, 2][[;;2]] == {1, 0}&] (* Jean-François Alcover, Oct 25 2021 *)

pxnm1(n,p) = { forprime(x=2,n, p1 = x; while(p1>1, if(p1%2==0,p1/=2,p1 = p1*p-1;); if(p1 == 4,break); ); if(p1 == 4,print1(x" ")) ) }

A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 17, 7, 11, 7, 5, 19, 13, 17, 11, 13, 11, 37, 29, 19, 13, 7, 53, 23, 67, 31, 23, 17, 17, 29, 97, 41, 71, 53, 37, 19, 19, 67, 31, 101, 43, 73, 59, 41, 23, 41, 37, 71, 59, 103, 47, 79, 61, 43, 29, 11, 43, 73, 131, 61, 107, 83, 131, 97, 47, 31

Offset: 1

Alois P. Heinz, Sep 20 2015

			Square array A(n,k) begins:
:  2,  2,  3,  17,  5,  13,   7,  17, ...
:  3,  5,  7,  19, 11,  53,  29,  67, ...
:  5, 11, 13,  37, 23,  97,  31,  71, ...
:  7, 17, 29,  67, 41, 101,  59, 131, ...
: 11, 19, 31,  71, 43, 103,  61, 137, ...
: 13, 23, 53,  73, 47, 107, 113, 139, ...
: 17, 37, 59,  79, 83, 109, 127, 257, ...
: 19, 41, 61, 131, 89, 193, 227, 263, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened
Index entries for primes involving decimal expansion of n

Columns k=1-7 give: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Row n=1 gives A164022.
Main diagonal gives A262366.
Cf. A262350, A262369.

u:= (h, t)-> select(isprime, [seq(h*2^t+k, k=0..2^t-1)]):
A:= proc(n, k) local l, p;
      l:= proc() [] end; p:= proc() -1 end;
      while nops(l(k))

nmax = 14;
col[k_] := col[k] = Module[{bk = IntegerDigits[k, 2], lk, pp = {}, coe = 1}, lbk = Length[bk]; While[Length[pp] < nmax, pp = Select[Prime[Range[ coe*nmax]], Quiet@Take[IntegerDigits[#, 2], lbk] == bk&]; coe++]; pp];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 25 2021 *)

A262284 Primes whose binary expansion begins 101.

Original entry on oeis.org

5, 11, 23, 41, 43, 47, 83, 89, 163, 167, 173, 179, 181, 191, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399

Offset: 1

N. J. A. Sloane, Sep 19 2015

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Suggested by A262350.
Column k=5 of A262365.

lis:=[]; q:=5;
for i from 1 to 10 do for j from 1 to 2^i-1 do
if isprime(q*2^i+j) then lis:=[op(lis),q*2^i+j]; fi; od: od:
lis;

Select[Flatten[Table[FromDigits[#,2]&/@(Join[{1,0,1},#]&/@Tuples[{0,1},n]),{n,0,10}]],PrimeQ] (* Harvey P. Dale, Oct 17 2021 *)

A262285 Primes whose binary expansion begins 111.

Original entry on oeis.org

7, 29, 31, 59, 61, 113, 127, 227, 229, 233, 239, 241, 251, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873

Offset: 1

N. J. A. Sloane, Sep 19 2015

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Suggested by A262350.
Column k=7 of A262365.

lis:=[]; q:=7;
for i from 1 to 10 do for j from 1 to 2^i-1 do
if isprime(q*2^i+j) then lis:=[op(lis),q*2^i+j]; fi; od: od:
lis;

Select[FromDigits[#,2]&/@(Join[{1,1,1},#]&/@Flatten[Table[Tuples[{0,1},n],{n,0,8}],1]),PrimeQ] (* or *) Select[Prime[Range[ 3,350]],Take[ IntegerDigits[ #,2],3]=={1,1,1}&] (* Harvey P. Dale, May 02 2021 *)

A262286 Primes whose binary expansion begins 100.

Original entry on oeis.org

17, 19, 37, 67, 71, 73, 79, 131, 137, 139, 149, 151, 157, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093

Offset: 1

N. J. A. Sloane, Sep 19 2015

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Suggested by A262350.
Column k=4 of A262365.

lis:=[]; q:=4;
for i from 1 to 10 do for j from 1 to 2^i-1 do
if isprime(q*2^i+j) then lis:=[op(lis),q*2^i+j]; fi; od: od:
lis;

A262287 Primes whose binary expansion begins 110.

Original entry on oeis.org

13, 53, 97, 101, 103, 107, 109, 193, 197, 199, 211, 223, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627

Offset: 1

N. J. A. Sloane, Sep 19 2015

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Suggested by A262350.
Column k=6 of A262365.

A262350 a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the leftmost 1 and following consecutive 0's removed. Then a(n) is the smallest prime not yet present whose binary representation begins with s.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 29, 53, 43, 23, 31, 61, 59, 109, 181, 107, 173, 367, 223, 191, 127, 509, 1013, 4013, 3931, 3767, 13757, 11131, 2939, 1783, 3037, 1979, 3821, 3547, 1499, 1901, 877, 2927, 1759, 1471, 1789, 1531, 2029, 2011, 7901, 60887, 56239, 93887, 28351

Offset: 1

Alois P. Heinz, Sep 18 2015

This sequence is infinite. The number of primes that are not in this sequence is conjectured to be infinite.

Proof of first statement, following a comment from David W. Wilson: It follows from standard results about primes in short intervals (see for example Harman, 1982) that there are infinitely many numbers in any base b starting with any nonzero prefix c. So there are infinitely many primes whose binary expansion begins with s, and so a(n) always exists. - N. J. A. Sloane, Sep 19 2015

			: 10                             ... 2
:   11                           ... 3
:    101                         ... 5
:      111                       ... 7
:       1101                     ... 13
:        1011                    ... 11
:          11101                 ... 29
:           110101               ... 53
:            101011              ... 43
:              10111             ... 23
:                11111           ... 31
:                 111101         ... 61
:                  111011        ... 59
:                   1101101      ... 109
:                    10110101    ... 181
:                      1101011   ... 107
:                       10101101 ... 173

Alois P. Heinz, Table of n, a(n) for n = 1..589
G. Harman, Primes in short intervals, Math. Zeit., 180 (1982), 335-348.

Binary analog of A262283.
Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Cf. A262365.

b:= proc() true end:
a:= proc(n) option remember; local h, k, ok, p, t;
      if n=1 then p:=2
    else h:= (k-> irem(k, 2^(ilog2(k))))(a(n-1)); p:= h;
         ok:= isprime(p) and b(p);
         for t while not ok do
           for k to 2^t-1 while not ok do p:= h*2^t+k;
             ok:= isprime(p) and b(p)
           od
         od
      fi; b(p):= false; p
    end:
seq(a(n), n=1..70);

A080168 Primes beginning and ending with '11' in binary representation.

Original entry on oeis.org

3, 7, 31, 59, 103, 107, 127, 199, 211, 223, 227, 239, 251, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 787, 811, 823, 827, 839, 859, 863, 883, 887, 907, 911, 919, 947, 967, 971, 983, 991, 1019, 1543, 1559, 1567, 1571, 1579, 1583, 1607, 1619, 1627

Offset: 1

Reinhard Zumkeller, Feb 04 2003

Intersection of A002145 and A080166.

			A000040(17)=59 -> '111011' therefore 59 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A004676, A074832, A030101.

Select[Prime[Range[300]],Take[IntegerDigits[#,2],2]==Take[ IntegerDigits[ #,2],-2]=={1,1}&] (* Harvey P. Dale, Jul 28 2021 *)

cp's OEIS Frontend

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

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A080165 Primes having initial digits "10" in binary representation.

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A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A262284 Primes whose binary expansion begins 101.

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A262285 Primes whose binary expansion begins 111.

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A262286 Primes whose binary expansion begins 100.

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A262287 Primes whose binary expansion begins 110.

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A262350 a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the leftmost 1 and following consecutive 0's removed. Then a(n) is the smallest prime not yet present whose binary representation begins with s.

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A080168 Primes beginning and ending with '11' in binary representation.

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