A065727 -id:A065727

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A065720

Primes whose binary representation is also the decimal representation of a prime.

+10
86

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 199, 229, 313, 331, 367, 379, 383, 443, 457, 523, 587, 631, 643, 647, 653, 659, 709, 883, 947, 997, 1009, 1091, 1097, 1163, 1259, 1277, 1283, 1289, 1321, 1483, 1601, 1669, 1693, 1709, 1753, 1877, 2063, 2069, 2099

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

In general rebase notation (Marc LeBrun): p2 = (2) [p] (10).

Also: Primes in A036952. - M. F. Hasler, Dec 11 2012

See A089971 for the binary representation of these terms. - M. F. Hasler, Jan 05 2014

LINKS

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

FORMULA

Equals A036952 intersect A000040. - M. F. Hasler, Dec 11 2012

EXAMPLE

1009{10} = 1111110001{2} is prime, and 1111110001{10} is also prime.

89 is in the sequence because it is a prime. Binary representation of 89 = 1011001, which is also a prime.

MAPLE

select(t -> isprime(t) and isprime(convert(t, binary)), [seq(2*i+1, i=1..1000)]); # Robert Israel, Jul 08 2014

MATHEMATICA

Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 2]]] & ]

Select[ Prime@ Range@ 330, PrimeQ[ FromDigits[ IntegerDigits[#, 2]]] &] (* Robert G. Wilson v, Oct 09 2014 *)

PROG

(PARI) {(baseE(x, b)= local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); e); n=0; for (m=1, 10^9, p=prime(m); b=baseE(p, 2); if (isprime(b), write("b065720.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 27 2009

(PARI) isok(p) = isprime(p) && isprime(fromdigits(binary(p), 10)); \\ Michel Marcus, Mar 04 2022

(Python)

from sympy import isprime

def ok(n): return isprime(n) and isprime(int(bin(n)[2:]))

print([k for k in range(2100) if ok(k)]) # Michael S. Branicky, Mar 04 2022

CROSSREFS

Cf. A065721, A065722, A065723, A065724, A065725, A065726, A065727, A065361.

Cf. A123266, A256621, A256622.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Nov 15 2001

EXTENSIONS

a(48)-a(50) from K. D. Bajpai, Jul 04 2014

STATUS

approved

A235395

Primes whose decimal representation is a valid number in base 9 and interpreted as such is again a prime.

+10
68

2, 3, 5, 7, 41, 47, 67, 131, 151, 241, 331, 337, 461, 557, 601, 641, 661, 751, 757, 827, 887, 1031, 1181, 1217, 1231, 1321, 1327, 1367, 1471, 1637, 1877, 2027, 2081, 2111, 2131, 2207, 2281, 2287, 2351, 2357, 2647, 2731, 2861, 3037, 3121, 3181, 3187, 3307, 3347

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

MATHEMATICA

Select[FromDigits@# & /@ IntegerDigits[ Prime@ Range@ 270, 9], PrimeQ]

PROG

(PARI) fixBase(n, oldBase, newBase)=my(d=digits(n, oldBase), t=newBase-1); for(i=1, #d, if(d[i]>t, for(j=i, #d, d[j]=t); break)); fromdigits(d, newBase)

list(lim)=my(v=List(), t); forprime(p=2, fixBase(lim\1, 10, 9), if(isprime(t=fromdigits(digits(p, 9), 10)), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Nov 07 2016

CROSSREFS

Cf. A065727, A031974, A089971, A089981, A090707, A090708, A090709, A090710, A235394, A000040.

KEYWORD

base,easy,nonn

AUTHOR

Robert G. Wilson v, Jan 09 2014

STATUS

approved

A235265

Primes whose base-3 representation also is the base-2 representation of a prime.

+10
66

3, 13, 31, 37, 271, 283, 733, 757, 769, 1009, 1093, 2281, 2467, 2521, 2551, 2917, 3001, 3037, 3163, 3169, 3187, 3271, 6673, 7321, 7573, 9001, 9103, 9733, 19801, 19963, 20011, 20443, 20521, 20533, 20749, 21871, 21961, 22123, 22639, 22717, 27253, 28711, 28759, 29173, 29191, 59077, 61483, 61507, 61561, 65701, 65881

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence and A235383 and A229037 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014

This sequence was motivated by work initiated by V.J. Pohjola's post to the SeqFan list, which led to a clarification of the definition and correction of some errors, in sequences A089971, A089981 and A090707 through A090721. These sequences use "rebasing" (terminology of A065361) from some base b to base 10. Sequences A065720 - A065727 follow the same idea but use rebasing in the other sense, from base 10 to base b. The observation that only (10,b) and (b,10) had been considered so far led to the definition of this and related sequences: In a systematic approach, it seems natural to start with the smallest possible pairs of different bases, (2,3) and (3,2), then (2 <-> 4), (3 <-> 4), (2 <-> 5), etc.

Among the two possibilities using the smallest possible bases, 2 and 3, the present one seems a little bit more interesting, among others because not every base-3 representation is a valid base-2 representation (in contrast to the opposite case). This is also a reason why the present sequence grows much faster than the partner sequence A235266.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Veikko Pohjola, A090709 Decimal primes whose decimal representation in base 6 is also prime, SeqFan list, Jan 03 2014.

EXAMPLE

3 = 10_3 and 10_2 = 2 is prime. 13 = 111_3 and 111_2 = 7 is prime.

MAPLE

N:= 1000: # to get the first N terms

count:= 0:

for i from 1 while count < N do

p2:= ithprime(i);

L:= convert(p2, base, 2);

p3:= add(3^(j-1)*L[j], j=1..nops(L));

if isprime(p3) then

count:= count+1;

A235265[count]:= p3;

od:

[seq(A235265[i], i=1..N)]; # Robert Israel, May 04 2014

MATHEMATICA

b32pQ[n_]:=Module[{idn3=IntegerDigits[n, 3]}, Max[idn3]<2&&PrimeQ[ FromDigits[ idn3, 2]]]; Select[Prime[Range[7000]], b32pQ] (* Harvey P. Dale, Apr 24 2015 *)

PROG

(PARI) is(p, b=2, c=3)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

(Python)

from sympy import isprime, nextprime

def agen(): # generator of terms

p = 2

while True:

p3 = sum(3**i for i, bi in enumerate(bin(p)[2:][::-1]) if bi=='1')

if isprime(p3):

yield p3

p = nextprime(p)

g = agen()

print([next(g) for n in range(1, 52)]) # Michael S. Branicky, Jan 16 2022

CROSSREFS

Subset of A077717.

Cf. A235266, A065720 and A036952, A065721 - A065727, A235394, A235395, A089971 and A020449, A089981, A090707 - A091924, A235461 - A235482. See M. F. Hasler's OEIS wiki page for further cross-references.

KEYWORD

nonn,base,nice

AUTHOR

M. F. Hasler, Jan 05 2014

STATUS

approved

A235461

Primes whose base-4 representation also is the base 2-representation of a prime.

+10
64

5, 17, 257, 277, 337, 1093, 1109, 1297, 1361, 4357, 5189, 16453, 16657, 16661, 17489, 17669, 17681, 17749, 21521, 21569, 21589, 65537, 65557, 65617, 65809, 66821, 70657, 70981, 70997, 81937, 82241, 83221, 83269, 86017, 86357, 87317, 263429, 263489, 267541, 278549

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=4, thus a subsequence of A077718 and therefore also of A000695, the Moser-de Bruijn sequence.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

5 = 11_4 and 11_2 = 3 are both prime, so 5 is a term.

17 = 101_4 and 101_2 = 5 are both prime, so 17 is a term.

PROG

(PARI) is(p, b=2, c=4)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

(Python)

from itertools import islice

from sympy import nextprime, isprime

def A235461_gen(): # generator of terms

p = 1

while (p:=nextprime(p)):

if isprime(m:=int(bin(p)[2:], 4)):

yield m

A235461_list = list(islice(A235461_gen(), 20)) # Chai Wah Wu, Aug 21 2023

CROSSREFS

Cf. A090707 - A091924, A235462 - A235482. See the LINK for further cross-references.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 11 2014

EXTENSIONS

a(37)-a(40) from Robert Price, Nov 01 2023

STATUS

approved

A235482

Primes whose base-5 representation is also the base-9 representation of a prime.

+10
63

2, 3, 7, 11, 17, 19, 37, 41, 61, 67, 71, 97, 109, 131, 139, 149, 151, 157, 167, 191, 197, 211, 251, 269, 281, 337, 349, 367, 401, 409, 439, 449, 457, 467, 487, 491, 499, 521, 557, 569, 607, 619, 631, 647, 661, 739, 761, 769, 821, 829, 887, 907, 941, 947, 967, 1009, 1019, 1031, 1061, 1069, 1087

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

A subsequence of A197636 and of course of A000040 ⊂ A015919.

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

41 = 131_5 and 131_9 = 109 are both prime, so 41 is a term.

MATHEMATICA

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 5], 9] &] (* Giovanni Resta, Sep 12 2019 *)

PROG

(PARI) is(p, b=9, c=5)=isprime(vector(#d=digits(p, c), i, b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

CROSSREFS

Cf. A235265, A235266, A235461 - A235481, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395. See the LINK for further cross-references.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 12 2014

STATUS

approved

A090721

Primes whose representation in base 1024 can be interpreted as a decimal prime.

+10
56

2, 3, 5, 7, 1031, 1033, 3079, 4099, 6151, 7177, 1048583, 1049603, 1050631, 1051649, 1053697, 1054723, 2099203, 2100227, 2101249, 2102273, 2102279, 2105347, 3148801, 3148807, 3149831, 3150857, 3151879, 3153923, 3153929, 4198409, 4200451, 5242883

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

PROG

(PARI) is(n)=if(!isprime(n), return(0)); my(s, t, b=1); while(n, t=n%1024; if(t>9, return(0)); s+=t*b; b*=10; n>>=10); s \\ Charles R Greathouse IV, Feb 07 2013

(PARI) v=List(); forprime(p=2, 1e3, d=digits(p); t=sum(i=1, #d, d[i]<<(10*(#d-i))); if(ispseudoprime(t), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Feb 07 2013

CROSSREFS

Cf. A089971, A089981, A090707 - A090720, A065720 - A065727.

KEYWORD

base,nonn

AUTHOR

Cino Hilliard, Jan 18 2004

EXTENSIONS

Better definition and offset by Omar E. Pol, Dec 24 2008

a(16) and a(26) corrected by Charles R Greathouse IV, Feb 07 2013

STATUS

approved

A065721

Primes p whose base-3 expansion is also the decimal expansion of a prime.

+10
41

2, 67, 79, 103, 139, 157, 181, 193, 199, 211, 229, 277, 283, 307, 313, 349, 367, 373, 409, 421, 433, 439, 463, 523, 541, 547, 571, 577, 751, 829, 883, 919, 1021, 1033, 1039, 1087, 1171, 1249, 1303, 1429, 1483, 1579, 1597, 1621, 1741, 1783, 1789, 1873

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

In general rebase notation (Marc LeBrun): p3 = (3) [p] (10).

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

1033_10 = 1102021_3 is prime, and so is 1102021_10.

MATHEMATICA

Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 3]]] & ]

PROG

(PARI) baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) } { n=0; for (m=1, 10^9, p=prime(m); b=baseE(p, 3); if (isprime(b), write("b065721.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 27 2009

(PARI) is(p, b=10, c=3)=isprime(vector(#c=digits(p, c), i, b^(#c-i))*c~)&&isprime(p) \\ M. F. Hasler, Jan 12 2014

CROSSREFS

Cf. A065720 up to A065727, A065361. See the Links for further cross-references.

KEYWORD

nonn,base

AUTHOR

Patrick De Geest, Nov 15 2001

EXTENSIONS

Definition clarified by M. F. Hasler, Jan 12 2014

STATUS

approved

A235615

Primes whose base-5 representation also is the base-4 representation of a prime.

+10
28

2, 3, 13, 41, 43, 61, 181, 191, 263, 281, 283, 331, 383, 431, 443, 463, 641, 643, 661, 881, 911, 1063, 1091, 1291, 1303, 1531, 1693, 2083, 2143, 2203, 2293, 2341, 3163, 3181, 3191, 3253, 3343, 3593, 3761, 3931, 4001, 4093, 4391, 4691, 4793, 5011, 5393, 5413, 5441, 6301

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

LINKS

Robert Price, Table of n, a(n) for n = 1..107268

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

Both 13 = 23_5 and 23_4 = 11 are prime.

PROG

(PARI) is(p, b=4, c=5)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

(PARI) forprime(p=1, 3e3, is(p, 5, 4)&&print1(vector(#d=digits(p, 4), i, 5^(#d-i))*d~, ", ")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(., 4, 5)

CROSSREFS

Cf. A235474, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 13 2014

STATUS

approved

A235639

Primes whose base-9 representation is also the base-6 representation of a prime.

+10
27

2, 3, 5, 19, 23, 41, 113, 127, 131, 163, 199, 271, 419, 433, 739, 743, 761, 919, 991, 1009, 1013, 1063, 1153, 1171, 1459, 1481, 1499, 1553, 1567, 1571, 1733, 1747, 1783, 1873, 1913, 2237, 2377, 2381, 2539, 2557, 2593, 2633, 2939, 3011, 3079, 3083, 3187, 3259, 3331, 3659

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

19 = 21_9 and 21_6 = 13 are both prime, so 19 is a term.

509 = 625_9 and 625_6 = 17 are both prime, but 625 is not a valid base-6 integer, so 509 is not a term.

MAPLE

R:= 2: x:= 2: count:= 1:

while count < 100 do

x:= nextprime(x);

L:= convert(x, base, 6);

y:= add(9^(i-1)*L[i], i=1..nops(L));

if isprime(y) then count:= count+1; R:= R, y fi

od:

R; # Robert Israel, May 18 2020

PROG

(PARI) is(p, b=6, c=9)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

(PARI) forprime(p=1, 3e3, is(p, 9, 6)&&print1(vector(#d=digits(p, 6), i, 9^(#d-i))*d~, ", ")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(., 6, 9)

CROSSREFS

Cf. A231481, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235638. See the LINK for further cross-references.

KEYWORD

nonn,base,look

AUTHOR

M. F. Hasler, Jan 13 2014

STATUS

approved

A235475

Primes whose base-2 representation also is the base-5 representation of a prime.

+10
9

2, 7, 11, 13, 19, 41, 59, 127, 151, 157, 167, 173, 181, 191, 223, 233, 241, 271, 313, 331, 409, 421, 443, 463, 541, 563, 577, 607, 613, 641, 701, 709, 733, 743, 809, 859, 877, 919, 929, 953, 967, 991, 1021, 1033, 1069, 1087, 1193, 1259, 1373, 1423, 1451, 1453, 1471, 1483, 1493, 1549, 1697, 1753, 1759, 1783, 1787, 1831, 1877, 1979, 1993

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

LINKS

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

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

7 = 111_2 and 111_5 = 31 are both prime, so 7 is a term.

MATHEMATICA

Select[Prime[Range[400]], PrimeQ[FromDigits[IntegerDigits[#, 2], 5]]&] (* Harvey P. Dale, Jun 15 2019 *)

PROG

(PARI) is(p, b=5, c=2)=isprime(vector(#d=digits(p, c), i, b^(#d-i))*d~)&&isprime(p) \\ This code is only valid for b>c.

CROSSREFS

Cf. A235266, A152079, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 12 2014

STATUS

approved

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