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

Author

Sean Oneil, Jun 30 2015

Keywords

• nonn
• base
• less

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)).

Examples

			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, ...
		

Links

• 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.

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.

Programs

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

Formula

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