A134028 -id:A134028 - cp's OEIS frontend

A323782 Prime numbers such that the reverse of the balanced ternary representation is a prime or a negated prime.

2, 5, 7, 11, 13, 17, 29, 31, 37, 43, 53, 59, 61, 71, 73, 83, 89, 101, 103, 137, 139, 149, 163, 173, 179, 181, 193, 199, 223, 233, 241, 263, 269, 277, 311, 313, 331, 347, 353, 367, 373, 379, 383, 389, 401, 421, 443, 449, 457, 467, 479, 487, 499, 509, 541

Offset: 1

Philippe Cochin, Jan 27 2019

The "warp" operation is an inverse map connecting this sequence and A323783.

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

Cf. A000040, A134028.
Supersequence of A224502.
Corresponding primes and -primes are in sequence A323783.
Primes that don't "warp" to a prime numbers are in sequence A323784.

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);
isok(n) = isprime(n) && isprime(abs(f(n))); \\ Michel Marcus, Jan 29 2019

is(n) = {if(!isprime(n), return(0)); my(d = digits(n, 3)); forstep(i = #d, 2, -1, if(d[i] >= 2, d[i] -= 3; d[i-1]++)); if(d[1] >= 2, d[1]-=3; d = concat(1, d)); isprime(abs(fromdigits(Vecrev(d), 3)))} \\ David A. Corneth, Feb 14 2019

Python
```
# See links.
```

A323784 Prime numbers such that the reverse of the balanced ternary representation is neither prime nor a negated prime.

Original entry on oeis.org

3, 19, 23, 41, 47, 67, 79, 97, 107, 109, 113, 127, 131, 151, 157, 167, 191, 197, 211, 227, 229, 239, 251, 257, 271, 281, 283, 293, 307, 317, 337, 349, 359, 397, 409, 419, 431, 433, 439, 461, 463, 491, 503, 521, 523, 557, 563, 571, 577, 587, 593, 617, 619, 631, 641, 647, 653, 659, 661, 673, 683, 701, 733, 743, 769, 787, 797

Offset: 1

Philippe Cochin, Jan 28 2019

			79 is a term:
79 is +00-+ in balanced ternary notation
+00-+ reversed is +-00+
+-00+ is 55 in balanced ternary notation
55 prime factors are 5 and 11
Therefore 55 is not prime.
Therefore the prime number 79 "warps" to the nonprime number 55.
This operation is reversible: 55 "warps" to 79.

Cf. A134028, A323782.

Complement of A323782 among primes.

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);
isok(n) = isprime(n) && !isprime(abs(f(n))); \\ Michel Marcus, Jan 29 2019

Python
```
# See Github link.
```

A327252 Balanced reversed ternary: Write n as ternary, reverse the order of the digits, then replace all 2's with (-1)'s.

Original entry on oeis.org

0, 1, -1, 1, 4, -2, -1, 2, -4, 1, 10, -8, 4, 13, -5, -2, 7, -11, -1, 8, -10, 2, 11, -7, -4, 5, -13, 1, 28, -26, 10, 37, -17, -8, 19, -35, 4, 31, -23, 13, 40, -14, -5, 22, -32, -2, 25, -29, 7, 34, -20, -11, 16, -38, -1, 26, -28, 8, 35, -19, -10, 17, -37, 2, 29, -25, 11

Offset: 0

Elisabeth Zemack, Sep 15 2019

			5 in base 3 is 12; reversing the ternary gives 21; replacing the 2 with (-1) gives (-1),1 which is -2 in decimal.

Rémy Sigrist, Table of n, a(n) for n = 0..19683

Cf. A134028 (reversed balanced ternary, i.e. balancing the ternary, then reversing the ternary and transforming back into decimal), A117966 (balanced ternary enumeration).

a:= proc(n) local m, r; m, r:= n, 0;
      while m>0 do m, r:= iquo(m, 3), 3*r+mods(m, 3) od; r
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Sep 17 2019

a[n_] := FromDigits[Reverse[IntegerDigits[n, 3]] /. {2 -> -1}, 3]; Array[a, 67, 0] (* Giovanni Resta, Sep 17 2019 *)

a(n) = fromdigits(apply(d -> if (d==2, -1, d), Vecrev(digits(n,3))), 3) \\ Rémy Sigrist, Sep 15 2019

def a(n):
    ternary = []
    i = math.floor(math.log(n,3))
    while i >= 0:
        for j in range (2, -1, -1):
            if j*(3**i) <= n:
                if (j==2): ternary.append(-1)
                else: ternary.append(j)
                n -= j*(3**i)
                break
        i -=1
    for k in range (0, len(ternary)):
        n += ternary[k]*(3**k)
    return n

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A323782 Prime numbers such that the reverse of the balanced ternary representation is a prime or a negated prime.

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A323784 Prime numbers such that the reverse of the balanced ternary representation is neither prime nor a negated prime.

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Author

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Examples

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PARI

Python

A327252 Balanced reversed ternary: Write n as ternary, reverse the order of the digits, then replace all 2's with (-1)'s.

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Maple

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Python