A379049 a(n) = prime(i)*dp(n,i) + prime(i)*dn(n,i) where dp(n,i) = 1 when the i-th trit of n is 1, dn(n,i) = 1 when the i-th trit of n is T, and dp(n,i) = dn(n,i) = 0 when the i-th trit of n is 0.

2, 3, 5, 4, 7, 11, 8, 13, 7, 6, 11, 17, 16, 31, 37, 22, 29, 17, 12, 19, 31, 26, 47, 13, 10, 17, 9, 8, 15, 23, 22, 43, 41, 38, 73, 37, 36, 71, 107, 106, 211, 221, 116, 127, 81, 46, 57, 103, 68, 101, 53, 32, 43, 25, 18, 29, 47, 40, 73, 97, 76, 131, 69, 62, 117

Offset: 0

Author

Lei Zhou, Dec 14 2024

Keywords

• new
• base
• easy
• nonn

Comments

The Balanced Ternary presentation of a number is a series of 1, 0, and T, where T represent -1. For example, 35 = 110T = 1 * 3^3 + 1* 3^2 + 0 * 3 - 1 = 27 + 9 + 0 - 1.

Conjecture: All positive integers greater than 1 appear in this sequence at least once.

Examples

			When n = 0, its BT presentation is 0, thus a(0) = 1 + 1 = 2;
When n = 1, its BT presentation is 1, the first prime is 2, thus a(1) = 2 + 1 = 3;
...
When n = 14, its BT presentation is 1TTT, thus prime 7 appears before the plus sign and primes 5, 3, and 2 appear in the term after the plus sign, a(14) = 7 + 5*3*2 = 37;
...
By the same rule, when n = 64, its BT presentation is 1T101, thus prime 11, 5, 2 appear before the plus sign and prime 7 appears in the term after the plus sign, a(64) = 11*5*2 + 7 = 117.
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A005812, A134022, A134023, A140267.

Programs

Mathematica

BTDigits[m_Integer, g_]:= Module[{n = m, d, sign, t = g}, If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n]; d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++]; While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign; t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
res = {}; Do[BT = BTDigits[i, {0}]; BTl = Length[BT]; f = 1; b = 1; Do[If[BT[[j]] == 1, f = f*Prime[BTl - j + 1]]; If[BT[[j]] == -1, b = b*Prime[BTl - j + 1]], {j, 1, BTl}];  d = f + b; AppendTo[res, d], {i, 0, 64}]; res

Python

from sympy import prime
def A140267(n): # see A140267
    return
def A379049(n):
    x,y,z = 1,1,str(A140267(n))[::-1]
    for i in range(len(z)):
        if z[i] == "1":
            x *= prime(i+1)
        if z[i] == "2":
            y *= prime(i+1)
    return x+y # John Tyler Rascoe, Feb 27 2025