A259566 Numbers following gaps in the sequence of base-3 numbers that don't contain 0.

1, 4, 7, 13, 16, 22, 25, 40, 43, 49, 52, 67, 70, 76, 79, 121, 124, 130, 133, 148, 151, 157, 160, 202, 205, 211, 214, 229, 232, 238, 241, 364, 367, 373, 376, 391, 394, 400, 403, 445, 448, 454, 457, 472, 475, 481, 484, 607, 610, 616, 619, 634, 637, 643, 646, 688, 691, 697, 700, 715, 718, 724, 727, 1093, 1096, 1102, 1105, 1120

Offset

1,2

Comments

Partial sums for the convergent modified harmonic series in base 3 excluding 0 = Sum of 1/a(n) + 1/(a(n) + 1) = Sum of (2*a(n) + 1)/(a(n)*(a(n) + 1)).

Links

Table of n, a(n) for n=1..68.

Robert Baillie, Sums of Reciprocals of Integers Missing a Given Digit, American Mathematical Monthly (Washington, DC: Mathematical Association of America) 86 (5): 372-374, May 1979, doi:10.2307/2321096. ISSN 0002-9890. JSTOR 2321096.

Formula

a(n) = A032924(2n - 1).

Example

Pattern of numbers of skipped terms (numbers in base 3 with at least one zero) is 1 (3 = 10_3), 1 (6 = 20_3), 3+1 (9 = 100_3, 10 = 101_3, 11 = 102_3, 12 = 110_3), 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 27+9+3+1, ...

Prog

(PARI) lista(nn)=prec0 = 1; for(n=1, nn, if (vecmin(digits(n, 3)), if (prec0, print1(n, , ", ")); prec0 = 0, prec0 = 1); ); \\ Michel Marcus, Aug 03 2015
(Python)
def A259566(n): return int(bin(m:=n)[3:], 3)*3 + (3**m.bit_length()-1>>1) if n>1 else 1 # Chai Wah Wu, Oct 13 2023

Crossrefs

Cf. A032924.

Subset of A016777 (congruent to 1 mod 3).

Each term is one more than each number that follows a gap in A081605.

Sequence in context: A310805 A310806 A125758 * A151788 A310807 A310808

Adjacent sequences: A259563 A259564 A259565 * A259567 A259568 A259569

Keyword

nonn,base,less

Author

Sean Oneil, Jun 30 2015

Status

approved