A001363 -id:A001363 - cp's OEIS frontend

A004678 Primes written in base 4.

2, 3, 11, 13, 23, 31, 101, 103, 113, 131, 133, 211, 221, 223, 233, 311, 323, 331, 1003, 1013, 1021, 1033, 1103, 1121, 1201, 1211, 1213, 1223, 1231, 1301, 1333, 2003, 2021, 2023, 2111, 2113, 2131, 2203, 2213, 2231, 2303, 2311, 2333, 3001, 3011, 3013

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Analogs in other bases: A004676 (base 2), A001363 (base 3), A004679 (base 5), A004680 (base 6), A004681 (base 7), A004682 (base 8), A004683 (base 9), A000040 (base 10), A004684 (base 11).

Cf. A072805 (primes of form 4k+3 written in base 3).

[Seqint(Intseq(NthPrime(n), 4)): n in [1..60]]; // G. C. Greubel, Oct 12 2018

FromDigits/@IntegerDigits[Prime[Range[50]],4] (* Vincenzo Librandi, Sep 03 2016 *)

a(n)=subst(Pol(digits(prime(n),4)),'x,10) \\ Charles R Greathouse IV, Nov 06 2013

a(n) = A007090(A000040(n)). - Jonathan Vos Post, Sep 09 2006

More terms from Vincenzo Librandi, Sep 03 2016

A004680 Primes written in base 6.

Original entry on oeis.org

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, 1011, 1015, 1021, 1025, 1035

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Analogs in other bases: A004676 (base 2), A001363 (base 3), A004678 (base 4), A004679 (base 5), A004681 (base 7), A004682 (base 8), A004683 (base 9), A000040 (base 10), A004684 (base 11).

Cf. A007092.

[Seqint(Intseq(NthPrime(n),6)): n in [1..60]]; // G. C. Greubel, Oct 10 2018

FromDigits/@IntegerDigits[Prime[Range[50]], 6] (* Vincenzo Librandi, Sep 03 2016 *)

a(n)=subst(Pol(digits(prime(n),6)),'x,10) \\ Charles R Greathouse IV, Nov 06 2013

vector(60, n, fromdigits(digits(prime(n), 6))) \\ G. C. Greubel, Oct 10 2018

a(n) = A007092(prime(n)). - Michel Marcus, Sep 03 2016

A340290 Numbers k that are the representation of primes in base 3 and in base 4.

Original entry on oeis.org

2, 1121, 2021, 2111, 10121, 10211, 11201, 12011, 12121, 12211, 21101, 21211, 22111, 101021, 101111, 110021, 110111, 110221, 111211, 112001, 121001, 121021, 122011, 200111, 201101, 210011, 211021, 211111, 222221, 1000211, 1002011, 1010111, 1011121, 1012201, 1021001

Offset: 1

Bernard Schott, Jan 03 2021

Except for a(1) = 2, which is the only even prime, all terms end with 1.
The corresponding sequences of primes are A235473 (for base 3) and A235467 (for base 4) (see examples).
As 1381 = 1220011_3 = 111211_4, prime 1381 occurs twice and is the next such prime after 2 (see example), which has a representation in base 3 and a representation in base 4 that are both terms of this sequence.

			a(1) = 2 and 2_3 = 2_4 = 2_10.
a(2) = 1121 because 1121_3 = 43_10 and 1121_4 = 89_10 are primes.
a(3) = 2021 because 2021_3 = 61_10 and 2021_4 = 137_10 are primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Intersection of A001363 and A004678.
Cf. A089981 (bases 3 and 10).
Cf. A235467, A235473.

f[n_] := Module[{d = IntegerDigits[n, 3]}, If[PrimeQ[FromDigits[d, 4]], FromDigits[d, 10], 0]]; seq = {}; Do[If[PrimeQ[n], m = f[n]; If[m > 0, AppendTo[seq, m]]], {n, 2, 1000}]; seq (* Amiram Eldar, Jan 03 2021 *)
FromDigits[#]&/@Select[Tuples[{0,1,2},7],PrimeQ[FromDigits[#,4]] && PrimeQ[ FromDigits[ #,3]]&] (* Harvey P. Dale, Dec 15 2021 *)

f(n, b) = fromdigits(digits(n, b));
my(vp=primes(700)); setintersect(apply(x->f(x,3), vp), apply(x->f(x,4), vp)) \\ Michel Marcus, Jan 04 2021

forprime(p=2, 10^3, my(t=digits(p,3)); if( isprime( fromdigits(t,4)), print1(fromdigits(t,10),", "))) \\ Joerg Arndt, Jan 04 2021

from sympy import prime, isprime
from sympy.ntheory.factor_ import digits
A340290_list = [int(s) for s in (''.join(str(d) for d in digits(prime(i),3)[1:]) for i in range(1,1000)) if isprime(int(s,4))] # Chai Wah Wu, Jan 09 2021

More terms from Amiram Eldar, Jan 03 2021

A178388 Concatenation of the first n primes written in base 3.

Original entry on oeis.org

2, 210, 21012, 2101221, 2101221102, 2101221102111, 2101221102111122, 2101221102111122201, 2101221102111122201212, 21012211021111222012121002, 210122110211112220121210021011, 2101221102111122201212100210111101

Offset: 1

Jonathan Vos Post, May 26 2010

			a(4) = Concatenate[prime(1) base 3, prime(2) base 3, prime(3) base 3, prime(3) base 3] = Concatenate[2 base 3, 3 base 3, 5 base 3, 7 base 3] = Concatenate[2, 10, 12, 21] = 2101221.

Rémy Sigrist, Table of n, a(n) for n = 1..174

Cf. A000040, A001363, A004676, A007088, A007089, A019518, A031974, A100003, A154703.

Module[{nn=15,p3},p3=IntegerDigits[Prime[Range[nn]],3];Table[FromDigits[Flatten[ Take[p3,n]]],{n,nn}]] (* Harvey P. Dale, Aug 25 2022 *)

v = 0; for (n=1, 12, d = digits(prime(n), 3); v = v*10^#d + fromdigits(d); print1 (v ", ")) \\ Rémy Sigrist, Aug 07 2017

Edited by N. J. A. Sloane, Jul 02 2017

A280997 Primes that have exactly 3 ones in both their binary and ternary expansions.

Original entry on oeis.org

13, 37, 41, 67, 97, 131, 193, 577, 1033, 1153, 2053, 4129, 8209, 18433, 32771, 32801, 32833, 65539, 133121, 525313, 557057, 1049089, 4194433, 167772161, 268435459

Offset: 1

K. D. Bajpai, Jan 12 2017

Sequence is likely to be finite. If it exists, a(26) > 10^200. - Robert Israel, Jan 12 2017

			37 is in the sequence because it is a prime and its binary expansion 100101 and ternary expansion 1101 both have exactly 3 ones.
131 is in the sequence because it is a prime and its binary expansion 10000011 and ternary expansion 11212 both have exactly 3 ones.

Cf. A000040, A001363, A007088, A014311, A066196.

Subset of A281004.

A:= NULL:
for a from 2 to 100 do
  for b from 1 to a-1 do
    p:= 2^a + 2^b + 1;
    if numboccur(1, convert(p,base,3)) = 3 and isprime(p) then
      A:= A, p
    fi
od od:
A; # Robert Israel, Jan 12 2017

Select[Prime[Range[500000]], Count[IntegerDigits[#, 3], 1] == Count[IntegerDigits[#, 2], 1] == 3 &]
Select[Prime[Range[300000]],DigitCount[#,2,1]==DigitCount[#,3,1]==3&] (* The program generates the first 23 terms of the sequence. *) (* Harvey P. Dale, Jul 20 2025 *)

cp's OEIS Frontend

A004678 Primes written in base 4.

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A004680 Primes written in base 6.

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A340290 Numbers k that are the representation of primes in base 3 and in base 4.

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A178388 Concatenation of the first n primes written in base 3.

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A280997 Primes that have exactly 3 ones in both their binary and ternary expansions.

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Maple

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