A081605 -id:A081605 - cp's OEIS frontend

A382417 Numbers with at least one zero in their base-8 representation.

0, 8, 16, 24, 32, 40, 48, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 80, 88, 96, 104, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 136, 144, 152, 160, 168, 176, 184, 192, 193, 194, 195, 196, 197, 198, 199, 200, 208, 216, 224, 232, 240, 248, 256, 257, 258, 259, 260

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382418 (base 9), A011540 (base 10).

Cf. A007094, A043421, A255805 (complement).

Select[Range[0, 300], DigitCount[#, 8, 0] > 0 &]

A382418 Numbers with at least one zero in their base-9 representation.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 99, 108, 117, 126, 135, 144, 153, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 180, 189, 198, 207, 216, 225, 234, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 261, 270, 279, 288, 297

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382417 (base 8), A011540 (base 10).

Cf. A007095, A043453, A255808 (complement).

Select[Range[0, 300], DigitCount[#, 9, 0] > 0 &]

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

Original entry on oeis.org

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

Sean Oneil, Jun 30 2015

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)).

			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, ...

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.

Cf. A032924.
Subset of A016777 (congruent to 1 mod 3).
Each term is one more than each number that follows a gap in A081605.

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

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

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

A371222 Product of digits of (n written in base 3) mod 3.

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ctibor O. Zizka, Mar 18 2024

a(A032924(n)) = 1 or 2. For n >= 1, a(A032924(n)) - 1 = A309953(A032924(n)) mod 3 - 1 = A010059(n+1).

			n = 5: 5_10 = 12_3 thus a(5) = 1*2 mod 3 = 2.
n = 8: 8_10 = 22_3 thus a(8) = 2*2 mod 3 = 1.

Cf. A007089, A010059, A032924, A081605, A309953, A371281 (base 10).

a[n_] := Mod[Times @@ IntegerDigits[n, 3], 3]; Array[a, 100, 0] (* Amiram Eldar, Mar 18 2024 *)

from functools import reduce
from sympy.ntheory import digits
def A371222(n): return reduce(lambda a,b: a*b%3,digits(n,3)[1:],1) # Chai Wah Wu, Mar 19 2024

a(n) = A309953(n) mod 3.

a(A081605(n)) = 0.

cp's OEIS Frontend

A382417 Numbers with at least one zero in their base-8 representation.

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A382418 Numbers with at least one zero in their base-9 representation.

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

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A371222 Product of digits of (n written in base 3) mod 3.

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