A146085 Positive integers a(n) such that for every integer m == 1 (mod 3), m >= 4, there exists a unique representation of m as a sum of the form a(l) + 3*a(s).

1, 4, 7, 28, 31, 34, 55, 58, 61, 244, 247, 250, 271, 274, 277, 298, 301, 304, 487, 490, 493, 514, 517, 520, 541, 544, 547, 2188, 2191, 2194, 2215, 2218, 2221, 2242, 2245, 2248, 2431, 2434, 2437, 2458, 2461, 2464, 2485, 2488, 2491, 2674, 2677, 2680, 2701, 2704, 2707, 2728, 2731, 2734

Offset: 1

Author

Vladimir Shevelev, Oct 27 2008

Keywords

• nonn

Comments

Theorem. An integer is in the sequence iff in its expansion on base 3 all digits at the k-th position from the end, k=3, 5, 7, ..., are zeros while the first digit from the end is 1. To get the decomposition of m==1(mod 3) as sum a(l)+3a(s), write m-3 as Sum b_j 3^j, then a(l) = 1 + Sum_{j odd} b_j 3^j.

Examples

			If m=46, then we have 46=1*3^0+2*3^2+1*3^3, thus a(l)=1+1*3^3=28 and the required decomposition is: 46=28+3*4, such that a(s)=4. We see that l=4, s=2, i.e. "index coordinates" of 46 are (4, 2). Thus we have a one-to-one map of integers m==1(mod 3), m>=4, to the positive lattice points on the plane.
		

Crossrefs

Cf. A145812, A145818.

Programs

PARI

isok(n) = {my(d=Vecrev(digits(n, 3)), k=3); while (k <= #d, if (d[k], return (0)); k += 2;); d[1] == 1;} \\ Michel Marcus, Dec 09 2018

Extensions

More terms from Michel Marcus, Dec 09 2018