A262365 -id:A262365 - cp's OEIS frontend

A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

11, 2, 13, 3, 23, 17, 41, 31, 29, 19, 5, 43, 37, 211, 101, 61, 53, 47, 307, 223, 103, 7, 67, 59, 401, 311, 227, 107, 83, 71, 601, 503, 409, 313, 229, 109, 97, 89, 73, 607, 509, 419, 317, 233, 113, 101, 907, 809, 79, 613, 521, 421, 331, 239, 127

Offset: 1

Alois P. Heinz, Sep 20 2015

			Square array A(n,k) begins:
:  11,   2,   3,  41,   5,  61,   7,  83, ...
:  13,  23,  31,  43,  53,  67,  71,  89, ...
:  17,  29,  37,  47,  59, 601,  73, 809, ...
:  19, 211, 307, 401, 503, 607,  79, 811, ...
: 101, 223, 311, 409, 509, 613, 701, 821, ...
: 103, 227, 313, 419, 521, 617, 709, 823, ...
: 107, 229, 317, 421, 523, 619, 719, 827, ...
: 109, 233, 331, 431, 541, 631, 727, 829, ...

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

Columns k=1-9 give: A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715.
Row n=1 gives A018800.
Main diagonal gives A077345.
Cf. A077344, A262365.

u:= (h, t)-> select(isprime, [seq(h*10^t+k, k=0..10^t-1)]):
A:= proc(n, k) local l, p;
      l:= proc() [] end; p:= proc() -1 end;
      while nops(l(k))

u[h_, t_] := Select[Table[h*10^t + k, {k, 0, 10^t - 1}], PrimeQ];
A[n_, k_] := Module[{l, p}, l[] = {}; p[] = -1; While[Length[l[k]] < n, p[k] = p[k]+1; l[k] = Join[l[k], u[k, p[k]]]]; l[k][[n]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

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" ")) ) }

A080166 Primes having initial digits "11" in binary representation.

Original entry on oeis.org

3, 7, 13, 29, 31, 53, 59, 61, 97, 101, 103, 107, 109, 113, 127, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 769, 773, 787, 797, 809, 811, 821

Offset: 1

Reinhard Zumkeller, Feb 03 2003

Also primes that terminate at 3,2,1 in the x-1 problem: Repeat, if x is even divide by 2 else subtract 1, until 3 is reached. - Cino Hilliard, Mar 27 2003

Or, primes in A004760. - Vladimir Shevelev, May 04 2009

			A000040(16)=53 -> '110101' therefore 53 is a term.

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

Cf. A004676, A080168.
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=3 of A262365.

Select[Prime[Range[200]],Take[IntegerDigits[#,2],2]=={1,1}&] (* Harvey P. Dale, Jul 30 2019 *)

pxnm1(n,p) = { forprime(x=2,n, p1 = x; while(p1>1, if(p1%2==0,p1/=2,p1 = p1*p-1;); if(p1 == 3,break); ); if(p1 == 3,print1(x" ")) ) }

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);

A164022 a(n) = the smallest prime that, when written in binary, starts with the substring of n in binary.

Original entry on oeis.org

2, 2, 3, 17, 5, 13, 7, 17, 19, 41, 11, 97, 13, 29, 31, 67, 17, 37, 19, 41, 43, 89, 23, 97, 101, 53, 109, 113, 29, 61, 31, 131, 67, 137, 71, 73, 37, 307, 79, 163, 41, 337, 43, 89, 181, 373, 47, 97, 197, 101, 103, 211, 53, 109, 223, 113, 229, 233, 59, 241, 61, 251, 127, 257

Offset: 1

Leroy Quet, Aug 08 2009

The argument used to prove that A018800(n) always exists applies here also. - N. J. A. Sloane, Nov 14 2014

			4 in binary is 100. Looking at the binary numbers that begin with 100: 100 = 4 in decimal is composite; 1000 = 8 in decimal is composite; 1001 = 9 in decimal is composite; 10000 = 16 in decimal is composite. But 10001 = 17 in decimal is prime. So a(4) = 17.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for primes involving decimal expansion of n

A018800 is the base-10 analog.

Row n=1 of A262365. Cf. A108234 (number of new bits), A208241 (proper substring).

A164022 := proc(n) dgs2 := convert(n,base,2) ; ldgs := nops(dgs2) ; for i from 1 do p := ithprime(i) ; if p >= n then pdgs := convert(p,base,2) ; if [op(nops(pdgs)+1-ldgs.. nops(pdgs),pdgs)] = dgs2 then RETURN( p) ; fi; fi; od: end: seq(A164022(n),n=1..120) ; # R. J. Mathar, Sep 13 2009

With[{s = Map[IntegerDigits[#, 2] &, Prime@ Range[10^4]]}, Table[Block[{d = IntegerDigits[n, 2]}, FromDigits[#, 2] &@ SelectFirst[s, Take[#, UpTo@ Length@ d] == d &]], {n, 64}]] (* Michael De Vlieger, Sep 23 2017 *)

Corrected terms a(1) and a(2) (with help from Ray Chandler) Leroy Quet, Aug 16 2009

Extended by R. J. Mathar, Sep 13 2009

A262366 a(n) is the n-th prime whose binary expansion begins with the binary expansion of n.

Original entry on oeis.org

2, 5, 13, 67, 43, 107, 127, 263, 307, 349, 373, 773, 839, 907, 991, 1063, 1109, 1201, 1277, 1321, 2713, 2819, 2963, 3119, 3229, 3371, 3517, 3691, 3779, 3943, 4051, 4217, 8461, 8719, 8963, 9241, 9497, 9767, 10039, 10303, 10613, 10799, 11159, 11317, 11657, 11923

Offset: 1

Alois P. Heinz, Sep 20 2015

Alois P. Heinz, Table of n, a(n) for n = 1..5000
Index entries for primes involving decimal expansion of n

Main diagonal of A262365.

Cf. A077345.

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)) A(n$2):
seq(a(n), n=1..60);

cp's OEIS Frontend

A262369 A(n,k) is the n-th prime whose decimal expansion begins with the decimal expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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

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A080166 Primes having initial digits "11" in binary representation.

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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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A164022 a(n) = the smallest prime that, when written in binary, starts with the substring of n in binary.

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