A036954 -id:A036954 - cp's OEIS frontend

A007089 Numbers in base 3.

0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1200, 1201, 1202, 1210, 1211

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Nonnegative integers with no decimal digit > 2. Thus nonnegative integers in base 10 whose quadrupling by normal addition or multiplication requires no carry operation. - Rick L. Shepherd, Jun 25 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.3 Positional Notation, p. 47.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
The Unicode Consortium, Tai Xuan Jing Symbols.
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.

Cf. A000042, A007088, A007090, A007091, A007092, A007093, A007094, A007095, A077267, A062756, A081603, A081604, A054635, A003137.

Primes when read as if in base 10: A036954.

a007089 0 = 0
a007089 n = 10 * a007089 n' + m where (n', m) = divMod n 3
-- Reinhard Zumkeller, Feb 19 2012

A007089 := proc(n) option remember;
if n <= 0 then 0
else
  if (n mod 3) = 0 then 10*procname(n/3) else procname(n-1) + 1 fi
fi end:
[seq(A007089(n), n=0..729)]; # - N. J. A. Sloane, Mar 09 2019

Table[ FromDigits[ IntegerDigits[n, 3]], {n, 0, 50}]

a(n)=if(n<1,0,if(n%3,a(n-1)+1,10*a(n/3)))

a(n)=fromdigits(digits(n,3)) \\ Charles R Greathouse IV, Jan 08 2017

def A007089(n):
  n,s = divmod(n,3); t = 1
  while n: n,r = divmod(n,3); t *= 10; s += r*t
  return s # M. F. Hasler, Feb 15 2023

a(0)=0, a(n) = 10*a(n/3) if n==0 (mod 3), a(n) = a(n-1) + 1 otherwise. - Benoit Cloitre, Dec 22 2002

a(n) = 10*a(floor(n/3)) + (n mod 3) if n > 0, a(0) = 0. - M. F. Hasler, Feb 15 2023

More terms from James Sellers, May 01 2000

A065721 Primes p whose base-3 expansion is also the decimal expansion of a prime.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Nov 15 2001

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

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

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

Primes in A036954.

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

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

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

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

A360502 Concatenate the ternary strings for 1,2,...,n.

Original entry on oeis.org

1, 12, 1210, 121011, 12101112, 1210111220, 121011122021, 12101112202122, 12101112202122100, 12101112202122100101, 12101112202122100101102, 12101112202122100101102110, 12101112202122100101102110111, 12101112202122100101102110111112, 12101112202122100101102110111112120

Offset: 1

N. J. A. Sloane, Feb 16 2023

If the terms are read as ternary strings and converted to base 10, we get A048435. For example, a(2) = 12_3 = 5_10, which is A048435(2). This is a prime, and gives the first term of A360503.
If the terms are read as decimal numbers, which of them are primes?  12101112202122100101102110111, for example, is not a prime, since it is 37*327057086543840543273030003.
When read as decimal numbers, the first prime is a(7315), with 56003 digits. - Michael S. Branicky, Apr 18 2023

			a(4): concatenate 1, 2, 10, 11, getting 121011.

Winston de Greef, Table of n, a(n) for n = 1..222
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A036954, A360503, A360504.

This is the ternary analog of A007908.

a:= proc(n) option remember; `if`(n=0, 0, (l-> parse(cat(
      a(n-1), seq(l[-i], i=1..nops(l)))))(convert(n, base, 3)))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 17 2023

nn = 15; s = IntegerDigits[Range[nn], 3]; Array[FromDigits[Join @@ s[[1 ;; #]]] &, nn] (* Michael De Vlieger, Apr 19 2023 *)

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(1, n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:=s+"".join(map(str, digits(n, 3)[1:]))) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

cp's OEIS Frontend

A007089 Numbers in base 3.

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A065721 Primes p whose base-3 expansion is also the decimal expansion of a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A360502 Concatenate the ternary strings for 1,2,...,n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python