A262350 -id:A262350 - cp's OEIS frontend

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.

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.

A262283 a(1)=2. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 31, 17, 71, 19, 97, 73, 37, 79, 907, 701, 101, 103, 307, 709, 911, 113, 131, 311, 1103, 1031, 313, 137, 373, 733, 331, 317, 173, 739, 397, 971, 719, 191, 919, 193, 937, 379, 797, 977, 773, 7307, 3079, 7901, 9011, 1109, 109, 929, 29, 941, 41

Offset: 1

N. J. A. Sloane, Sep 18 2015

If a(n-1) has a single digit then a(n) is simply the smallest missing prime.
Leading zeros in s are ignored.
The sequence is infinite, since there infinitely many primes that start with s (see the comments in A080165).
The data in the b-file suggests that there are infinitely many primes that do not appear. Hoever, at present that is no proof that even one prime (23, say) never appears. - N. J. A. Sloane, Sep 20 2015
Alois P. Heinz points out that a(n) = A262282(n+29) starting at the 103rd term. - N. J. A. Sloane, Sep 19 2015

			a(1)=2, so s is the empty string, so a(2) is the smallest missing prime, 3. After a(6)=13, s=3, so a(7) is the smallest missing prime that starts with 3, which is 31.

Alois P. Heinz, Table of n, a(n) for n = 1..676

Cf. A080165, A089755, A262282, A262350.

import Data.List (isPrefixOf, delete)
a262283 n = a262283_list !! (n-1)
a262283_list = 2 : f "" (map show $ tail a000040_list) where
   f xs pss = (read ys :: Integer) :
              f (dropWhile (== '0') ys') (delete ys pss)
              where ys@(_:ys') = head $ filter (isPrefixOf xs) pss
-- Reinhard Zumkeller, Sep 19 2015

More terms from Alois P. Heinz, Sep 18 2015

cp's OEIS Frontend

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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A262283 a(1)=2. For n>1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest prime not yet present which starts with s.

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