A134028 -id:A134028 - cp's OEIS frontend

A323783 a(n) = A134028(A323782(n)): Primes and negated primes such that the reverse of the balanced ternary representation is a prime.

-2, -11, 7, -5, 13, -29, -17, 37, 31, 43, -83, -101, 61, -89, 73, -53, -71, -59, 103, -173, 313, -353, 241, -137, -263, 223, 331, 277, 181, -269, 163, -179, -233, 199, -347, 139, 193, -311, -149, 367, 853, 691, -929, -443, -983, 421, -389, -839, 457, -677

Offset: 1

Philippe Cochin, Jan 27 2019

The "warp" operation is reversible between A323782 and this sequence.

Negating a number in balanced ternary notation is done by inverting the + and -.

			-17 is a term:
-17 is -+0+ in balanced ternary notation
-+0+ reversed is +0+-
+0+- is 29 in balanced ternary notation
29 is prime
Therefore -17 is "warped" to 29.
This operation is reversible: 29 "warps" to -17.

Corresponding warp prime numbers to A323782.

Supersequence of A224502.

d3(n) = if ((n%3)==2, n\3+1, n\3);
m3(n) = if ((n%3)==2, -1, n % 3);
t(n) = if (n==0, [0], if (abs(n) == 1, [n], concat(m3(n), t(d3(n)))));
f(n) = subst(Pol(Vec(t(n))), x, 3);
lista(nn) = {forprime(n=1, nn, if (isprime(abs(f(n))), print1(f(n), ", ")););} \\ Michel Marcus, Jan 29 2019

Python
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# See Github link.
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A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

Original entry on oeis.org

0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, -8, -10, -6, -5, -7, -12, -11, -13, 27, 28, 26, 30, 31, 29, 24, 25, 23, 36, 37, 35, 39, 40, 38, 33, 34, 32, 18, 19, 17, 21, 22, 20, 15, 16, 14, -27, -26, -28, -24, -23, -25, -30, -29, -31, -18, -17, -19, -15, -14, -16, -21, -20, -22, -36

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

As the graph demonstrates, there are large discontinuities in the sequence between terms 3^i-1 and 3^i, and between terms 2*3^i-1 and 2*3^i. - N. J. A. Sloane, Jul 03 2016

			7 in base 3 is 21; changing the 2 to a (-1) gives (-1)*3+1 = -2, so a(7) = -2. I.e., the number of -2 according to the balanced ternary enumeration is 7, which can be obtained by replacing every -1 by 2 in the balanced ternary representation (or expansion) of -2, which is -1,1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 729 terms from A. Karttunen)
Ken Levasseur, The Balanced Ternary Number System
Tilman Piesk, First 27 numbers with their ternary representation
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Wikipedia, Balanced ternary

Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.

Column k=1 of A319047.

f:= proc(n) local L,i;
   L:= subs(2=-1,convert(n,base,3));
   add(L[i]*3^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]);
# alternate:
N:= 100: # to get a(0) to a(N)
g:= 0:
for n from 1 to ceil(log[3](N+1)) do
g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
od:
seq(coeff(g,x,j),j=0..N); # Robert Israel, Nov 17 2015
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Aug 14 2019

Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* Michael De Vlieger, Nov 17 2015 *)

a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
vector(73, i, a(i-1))  \\ Gheorghe Coserea, Nov 17 2015

def a(n):
    if n==0: return 0
    if n%3==0: return 3*a(n//3)
    elif n%3==1: return 3*a((n - 1)//3) + 1
    else: return 3*a((n - 2)//3) - 1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 06 2017

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
G.f. satisfies A(x) = 3*A(x^3)*(1+x+x^2) + x/(1+x+x^2). - corrected by Robert Israel, Nov 17 2015
A004488(n) = a(n)^{-1}(-a(n)). I.e., if a(n) <= 0, A004488(n) = A117967(-a(n)) and if a(n) > 0, A004488(n) = A117968(a(n)).
a(n) = n - 3 * A005836(A289814(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

Name corrected by Andrey Zabolotskiy, Nov 10 2019

A030102 Base-3 reversal of n (written in base 10).

Original entry on oeis.org

0, 1, 2, 1, 4, 7, 2, 5, 8, 1, 10, 19, 4, 13, 22, 7, 16, 25, 2, 11, 20, 5, 14, 23, 8, 17, 26, 1, 28, 55, 10, 37, 64, 19, 46, 73, 4, 31, 58, 13, 40, 67, 22, 49, 76, 7, 34, 61, 16, 43, 70, 25, 52, 79, 2, 29, 56, 11, 38, 65, 20, 47, 74, 5, 32, 59, 14, 41, 68, 23, 50, 77, 8, 35, 62, 17, 44, 71

Offset: 0

David W. Wilson

			a(17) = 25 because 17 in base 3 is 122, and backwards that is 221, which is 25 in base 10.
a(18) = 2 because 18 in base 3 is 200, and backwards that is 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Lukas Spiegelhofer, A digit reversal property for an analogue of Stern's sequence, arXiv:1709.05651 [math.NT], 2017. See Theorem 1.1.

Cf. A134028, A030101 - A030108, A004086, A134028.
Cf. A030341.
Cf. A263273 for a bijective variant.

a030102 = foldl (\v d -> 3 * v + d) 0 . a030341_row
-- Reinhard Zumkeller, Dec 16 2013

a030102:= proc(n) option remember;
  local y;
  y:= n mod 3;
  3^ilog[3](n)*y + procname((n-y)/3)
end proc:
for i from 0 to 2 do a030102(i):= i od:
seq(a030102(i),i=0..100); # Robert Israel, Dec 24 2015
# alternative
A030102 := proc(n)
    local r ;
    r := ListTools[Reverse](convert(n,base,3)) ;
    add(op(i,r)*3^(i-1),i=1..nops(r)) ;
end proc: # R. J. Mathar, May 28 2016

A030102[n_] := FromDigits[Reverse@IntegerDigits[n, 3], 3] (* JungHwan Min, Dec 23 2015 *)
FromDigits[#,3]&/@(Reverse/@IntegerDigits[Range[0,80],3]) (* Harvey P. Dale, Feb 05 2020 *)

a(n,b=3)=subst(Polrev(base(n,b)),x,b) /* where */
base(n,b)={my(a=[n%b]);while(0M. F. Hasler, Nov 04 2011

a(n) = fromdigits(Vecrev(digits(n, 3)), 3); \\ Michel Marcus, Oct 10 2017

a(n) = t(n,0) with t(n,r) = if n=0 then r else t(floor(n/3),r*3+(n mod 3)). - Reinhard Zumkeller, Mar 04 2010

G.f. G(x) satisfies: G(x) = (1+x+x^2)*G(x^3) - (1+2*x)*(x + 2*Sum_{m>=0} 3^m*x^(3^(m+1)+1)/(x^3-1). - Robert Israel, Dec 24 2015

A134021 Length of n in balanced ternary representation.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Oct 19 2007

Shifted variant of A064099.

			100 = 1*3^4+1*3^3-1*3^2+0*3^1+1*3^0: a(100) = |++-0+| = 5.
200 = 1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0: a(200) = |+-+++-| = 6.
300 = 1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0: a(300) = |++-0+0| = 6.

Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Balanced Ternary.

Cf. A005812, A059095, A081604, A134022, A134023, A134024, A134025, A134026, A134027, A134028, A134421.

a[n_] := Ceiling[Log[3, 2*n+1]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Apr 03 2025 *)

def a(n):
    if n==0: return 1
    s=0
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

For n > 0: a(n) = ceiling(log(2*n+1)/log(3)).
a(n) = A134022(n) + A134023(n) + A134024(n).
0 <= a(n) - A081604(n) <= 1.
a(A134025(n)) = A081604(A134025(n)); a(A134026(n)) = A081604(A134026(n))+1.
a(A134027(n)) = a(n); a(abs(A134028(n))) <= a(n).
a(n) = A064099(n-1) for n>1.
n = Sum_{k=0..a(n)-1} (A059095(A134421(n)-2-k)*3^k), for n > 0. - Reinhard Zumkeller, Oct 25 2007
a(n) = A005812(n) + A134023(n).

A134027 Nonnegative numbers that are palindromes in balanced ternary representation.

Original entry on oeis.org

0, 1, 4, 7, 10, 13, 16, 28, 40, 43, 52, 61, 73, 82, 91, 103, 112, 121, 124, 160, 196, 208, 244, 280, 292, 328, 364, 367, 394, 421, 457, 484, 511, 547, 574, 601, 613, 640, 667, 703, 730, 757, 793, 820, 847, 859, 886, 913, 949, 976, 1003, 1039, 1066, 1093, 1096

Offset: 1

Reinhard Zumkeller, Oct 19 2007

A134028(a(n)) = a(n).

			a(10) = 43 = 1*3^4 - 1*3^3 - 1*3^2 - 1*3^1 + 1*3^0 == '+---+';
a(11) = 52 = 1*3^4 - 1*3^3 + 0*3^2 - 1*3^1 + 1*3^0 == '+-0-+';
a(12) = 61 = 1*3^4 - 1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 == '+-+-+';
a(13) = 73 = 1*3^4 + 0*3^3 - 1*3^2 + 0*3^1 + 1*3^0 == '+0-0+'.

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

Lei Zhou, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Wikipedia, Balanced Ternary

Cf. A014190.

balTernDigits[0] := {0}; balTernDigits[n_ /; n > 0] := Module[{unParsed = n, currRem, currExp = 1, digitList = {}, nextDigit}, While[unParsed > 0, If[unParsed == 3^(currExp - 1), digitList = Append[digitList, 1]; unParsed = 0, currRem = Mod[unParsed, 3^currExp]/3^(currExp - 1); nextDigit = Switch[ currRem, 0, 0, 2, -1, 1, 1]; digitList = Append[ digitList, nextDigit]; unParsed = unParsed - nextDigit*3^(currExp - 1)]; currExp++]; digitList = Reverse[digitList]; Return[ digitList]]; balTernDigits[n_ /; n < 0] := (-1) balTernDigits[ Abs[ n]]; palQ[n_] := n == Reverse@ n; Select[ Range@ 1300, palQ@ balTernDigits@# &] (* Robert G. Wilson v, Jun 17 2014 *)

A030104 Base 5 reversal of n (written in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 1, 26, 51, 76, 101, 6, 31, 56, 81, 106, 11, 36, 61, 86, 111, 16, 41, 66, 91, 116, 21, 46, 71, 96, 121, 2, 27, 52, 77, 102, 7, 32, 57, 82, 107, 12, 37, 62, 87, 112, 17, 42, 67, 92, 117, 22, 47, 72, 97

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..3124

Cf. A134028, A030101 - A030108, A004086, A055950, A319722.

IntegerReverse[Range[0, 100], 5] (* Paolo Xausa, Aug 07 2024 *)

a(n,b=5)=subst(Polrev(base(n,b)),x,b) /* where */
base(n,b)={my(a=[n%b]);while(n\=b,a=concat(n%b,a));a}  \\ M. F. Hasler, Nov 04 2011

A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 11, 6, 9, 12, 7, 10, 13, 14, 23, 32, 17, 26, 35, 20, 29, 38, 15, 24, 33, 18, 27, 36, 21, 30, 39, 16, 25, 34, 19, 28, 37, 22, 31, 40, 41, 68, 95, 50, 77, 104, 59, 86, 113, 44, 71, 98, 53, 80, 107, 62, 89, 116, 47, 74, 101, 56, 83, 110, 65, 92

Offset: 0

Kevin Ryde, Feb 19 2022

Self-inverse permutation with swaps confined to terms of a given digit length (A134021) so within blocks n = (3^k+1)/2 .. (3^(k+1)-1)/2.
Can extend to negative n by a(-n) = -a(n).
A072998 is balanced ternary coded in decimal digits so that reversal except first digit of A072998(n) is at A072998(a(n)).  Similarly its ternary equivalent A157671, and also A132141 ternary starting with 1.
These sequences all have a fixed initial digit followed by all ternary strings which is the reversed part.  A007932 is such strings as decimal digits 1,2,3 but it omits the empty string so the whole reversal of A007932(n) is at A007932(a(n+1)-1).
Fixed points a(n) = n are where n in balanced ternary is a palindrome apart from its initial 1.  These are the full balanced ternary palindromes with their least significant 1 removed, so all n = (A134027(m)-1)/3 for m>=2.

			n    = 224 = balanced ternary 1,  0, -1, 1,  0, -1
                      reverse     ^^^^^^^^^^^^^^^^
a(n) = 168 = balanced ternary 1, -1,  0, 1, -1,  0

Cf. A059095 (balanced ternary), A134028 (full reverse), A134027 (palindromes).
Cf. A072998, A157671, A132141, A007932.
In other bases: A059893 (binary), A343150 (Zeckendorf), A343152 (lazy Fibonacci).

a(n) = if(n==0,0, my(k=if(n,logint(n<<1,3)), s=(3^k+1)>>1); s + fromdigits(Vec(Vecrev(digits(n-s,3)),k),3));

A274107 (Balanced ternary representation of n) - (reversed balanced ternary representation of n).

Original entry on oeis.org

0, 0, 1, 2, 0, 13, -1, -9, 4, 8, 0, 13, 8, 0, 49, 17, -9, 34, -7, -33, 10, -13, -39, 28, -4, -30, 13, 26, 0, 43, 20, -6, 61, 29, 3, 46, 32, 6, 49, 26, 0, 157, 71, -9, 124, 29, -51, 82, 5, -75, 130, 44, -36, 97, -25, -105, 28, -49, -129, 76, -10, -90, 43, -25, -105, 28, -49, -129, 100, 14, -66

Offset: 0

N. J. A. Sloane, Jun 16 2016

This is A117966(n) - A134028(n).

As the graph shows, there are large jumps at places where either A117966 or A134028 has a discontinuity, that is, at values of n which are a power of 3 times 1, 1.5, or 2. - N. J. A. Sloane, Jul 03 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A117966, A134028.

A160652 Express n in balanced ternary, then reverse the digits, leaving any trailing zeros alone.

Original entry on oeis.org

0, 1, -2, 3, 4, -11, -6, 7, -8, 9, 10, -5, 12, 13, -38, -33, 16, -29, -18, 25, -20, 21, 34, -35, -24, 19, -26, 27, 28, -17, 30, 37, -32, -15, 22, -23, 36, 31, -14, 39, 40, -119, -114, 43, -92, -99, 70, -65, 48, 97, -110, -87, 52, -83, -54, 79, -56, 75, 106, -101

Offset: 0

Franklin T. Adams-Watters, May 21 2009

This sequence, together with its negative extension a(-n) = -a(n) is a self-inverse permutation of the integers. The absolute values are a self-inverse permutation of the nonnegative integers.

			87 in balanced ternary is 101(-1)0; leaving the final 0 and reversing the remaining digits gives (-1)1010, which is -51; so a(87) = -51.

Indranil Ghosh, Table of n, a(n) for n = 0..6561

Cf. A065363, A117962, A117966.

Cf. A057889, A134028, A007949. [Franklin T. Adams-Watters, May 24 2009]

a(n)=local(r,dr,q);if(n==0,0,r=0;dr=1;while(n%3==0,dr*=3;n\=3);while(n!=0,q=(n+1)\3;r=3*r+dr*(n-3*q);n=q);r) \\ Franklin T. Adams-Watters, May 24 2009

def a(n):
    if n==0: return 0
    r=0
    dr=1
    while n%3==0:
        dr*=3
        n/=3
    while n!=0:
        q=(n + 1)/3
        r=3*r + dr*(n - 3*q)
        n=q
    return r
##print [a(n) for n in range(101)] # Indranil Ghosh, Jun 10 2017, after Franklin T. Adams-Watters

a(n) = A134028(n)*3^A007949(n). [Franklin T. Adams-Watters, May 24 2009]

A274601 a(n) = 2*3^(s-1) - n, where s is the number of trits of n in balanced ternary form.

Original entry on oeis.org

1, 4, 3, 2, 13, 12, 11, 10, 9, 8, 7, 6, 5, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 121, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100

Offset: 1

Lei Zhou, Nov 10 2016

Analogous to a bit, a ternary digit is a trit.

Per the definition, n + a(n) = 2*3^(s-1), where s is the number of trits of n and a(n), n and a(n) form a decomposition of 2*3^(s-1).

			For n=1 the balanced ternary form of 1 is 1, which has 1 trits. 2*3^(1-1)-1 = 1, so a(1) = 1.
For n=2 the balanced ternary form of 2 is 1T, which has 2 trits. 2*3^(2-1)-2 = 4, so a(2) = 4.
For n=3 the balanced ternary form of 3 is 10, which has 2 trits. 2*3^(2-1)-3 = 3, so a(2) = 3.
...
For n=62 the balanced ternary form of 62 is 1T10T, which has 5 trits. 2*3^(5-1)-62 = 100, so a(62) = 100.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A117966, A134021, A140267, A134028.

Table[2*3^(Floor[Log[3, 2*n - 1]]) - n, {n, 1, 62}]

a(n) = 2*3^logint(2*n-1,3)-n \\ Jason Yuen, Nov 18 2024

from sympy import integer_log
def a(n): return 2*3**(integer_log(2*n - 1, 3)[0]) - n
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 10 2017

a(n) = 2*3^(floor(log_3(2*n-1))) - n. [corrected by Jason Yuen, Nov 18 2024]

cp's OEIS Frontend

A323783 a(n) = A134028(A323782(n)): Primes and negated primes such that the reverse of the balanced ternary representation is a prime.

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A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

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A030102 Base-3 reversal of n (written in base 10).

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A134021 Length of n in balanced ternary representation.

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A134027 Nonnegative numbers that are palindromes in balanced ternary representation.

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A030104 Base 5 reversal of n (written in base 10).

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A351702 In the balanced ternary representation of n, reverse the order of digits other than the most significant.

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