A047253 -id:A047253 - cp's OEIS frontend

A122841 Greatest k such that 6^k divides n.

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

Offset: 1

Reinhard Zumkeller, Sep 13 2006

nonn

See A054895 for the partial sums. - Hieronymus Fischer, Jun 08 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A122840, A007814, A007949, A054895, A234959.

a122841 = f 0 where
   f y x = if r > 0 then y else f (y + 1) x'
           where (x', r) = divMod x 6
-- Reinhard Zumkeller, Nov 10 2013

Table[IntegerExponent[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n) = valuation(n, 6); \\ Michel Marcus, Jan 17 2022

From Hieronymus Fischer, Jun 03 2012: (Start)
With m = floor(log_6(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/6^j))).
a(n) = m + Sum_{j=1..m} (floor(-frac(n/6^j))).
a(n) = A054895(n) - A054895(n-1).
G.f.: Sum_{j>0} x^6^j/(1-x^6^j). (End)
a(A047253(n)) = 0; a(A008588(n)) > 0; a(A044102(n)) > 1. - Reinhard Zumkeller, Nov 10 2013
6^a(n) = A234959(n), n >= 1. - Wolfdieter Lang, Jun 30 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/5. - Amiram Eldar, Jan 17 2022
a(n) = min(A007814(n), A007949(n)). - Jianing Song, Jul 23 2022

A097325 Period 6: repeat [0, 1, 1, 1, 1, 1].

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Ralf Stephan, Aug 16 2004

a(n) is 0 if 6 divides n, 1 otherwise.

Antti Karttunen, Table of n, a(n) for n = 0..26244
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Characteristic sequence of A047253.
Binary complement of A079979.
Cf. A010875, A168185, A145568, A168184, A168182, A168181, A109720, A011558, A166486, A011655, A000035.

[Sign(n mod 6) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(signum(k mod 6), k=0..100); # Wesley Ivan Hurt, Jun 29 2013

Table[Boole[Not[Divisible[n, 6]]], {n, 0, 89}] (* Alonso del Arte, Oct 21 2013 *)
PadRight[{}, 120, {0, 1, 1, 1, 1, 1}] (* Michael De Vlieger, Dec 22 2017 *)

PARI
```
a(n) = sign(n%6);
```

(define (A097325 n) (if (zero? (modulo n 6)) 0 1)) ;; Antti Karttunen, Dec 22 2017

G.f.: 1/(1-x) - 1/(1-x^6) = Sum_{k>=0} x^k - x^(6*k).
Recurrence: a(n+6) = a(n), a(0) = 0, a(i) = 1, 1 <= i <= 5.
a(n) = (1/4) * (3 - (-1)^n - (-1)^((n+1)/3) - (-1)^((2n+1)/3)).
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079979(n).
a(A047253(n)) = 1, a(A008588(n)) = 0.
A033438(n) = Sum_{k=0..n} a(k)*(n-k). (End)
Dirichlet g.f.: (1 - 1/6^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m, n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 6). - Wesley Ivan Hurt, Jun 29 2013
a(n) = ceiling(5n/6) - floor(5n/6). - Wesley Ivan Hurt, Jun 20 2014

New name from Omar E. Pol, Oct 21 2013

A248910 Numbers with no zeros in base-6 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A039215, A047253, A184522, A187390, A194386.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Senary
Index entries for 6-automatic sequences.

Cf. A007092, A100969 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A255805 (base 8), A255808 (base 9), A052382 (base 10).

a248910 n = a248910_list !! (n-1)
a248910_list = iterate f 1 where
   f x = 1 + if r < 5 then x else 6 * f x'  where (x', r) = divMod x 6

Select[Range[100], DigitCount[#,6, 0] == 0 &] (* Paolo Xausa, Jun 29 2025 *)

isok(m) = vecmin(digits(m, 6)) > 0; \\ Michel Marcus, Jan 23 2022

from sympy import integer_log
def A248910(n):
    m = integer_log(k:=(n<<2)+1,5)[0]
    return sum((1+(k-5**m)//(5**j<<2)%5)*6**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A108120 Floor[n*1/Sin[1]], or Beatty sequence for 1/sin(1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Zak Seidov, Jun 04 2005

nonn

Complement of A108587; not the same as A108586: a(37)=43 <> A108586(37)=44. - Reinhard Zumkeller, Jun 11 2005

Cf. A023800, A031943, A037918, A039215, A043091, A047253.

Cf. A108593.

a[n_]:=Floor[n*1/Sin[1]];Table[a[n], {n, 90}]

a(n) = floor(n*1/sin(1))

A367455 Numbers not divisible by 6 that are neither squarefree nor prime powers.

Original entry on oeis.org

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 88, 92, 98, 99, 100, 104, 112, 116, 117, 124, 135, 136, 140, 147, 148, 152, 153, 160, 164, 171, 172, 175, 176, 184, 188, 189, 196, 200, 207, 208, 212, 220, 224, 225, 232, 236, 242, 244, 245, 248, 250, 260, 261

Offset: 1

Michael De Vlieger, Jan 15 2024

A364997 is a proper subset.

The asymptotic density of this sequence is 1/6 - 1/(2*Pi^2). - Amiram Eldar, Jan 20 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A047253, A053669, A119288, A126706, A364997, A365710, A367018.

Select[Range[261], And[Nor[SquareFreeQ[#], PrimePowerQ[#]], Mod[#, 6] != 0] &]

Intersection of A047253 and A126706.
Let p = A119288(k) and q = A053669(k) for k in A126706. Various definitions of this sequence:
{a(n)} = { k : Omega(k) > omega(k) > 1, p > q }.
{a(n)} = { k : Omega(k) > omega(k) > 1, k mod 6 != 0 }.
{a(n)} = { k = mx : x in A367018, rad(m) | x, m > 1. }.

A232100 a(n) = the position of A232099(n) in A055926.

Original entry on oeis.org

41, 123, 205, 287, 369, 451, 533, 615, 697, 779, 861, 943, 1025, 1107, 1189, 1271, 1353, 1435, 1517, 1599, 1681, 1763, 1845, 1927, 2009, 2091, 2173, 2255, 2337, 2419, 2501, 2583, 2665, 2706, 2747, 2829, 2911, 2993, 3075, 3157, 3239, 3321, 3403, 3485, 3567, 3649

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

In range n=1..2215, each a(n) is a multiple of 41, with a(2215) = 177079 = 4319*41. In that range, all 2160 odd multiples 1, 3, 5, ..., 4319*41 occur, but only 55 even multiples, with a(34) = 66*41 = 2706 being the first one of them. The multipliers for even terms in that range seems to be given by 66*A047253(1..55) from 66*1 up to 66*A047253(55) = 66*65 = 4290. (Where A047253 gives the numbers not divisible by six).

Antti Karttunen, Table of n, a(n) for n = 1..10007

Cf. A232099, A055926, A047253, A232741.

A055926(a(n)) = A232099(n) for all n.

A108611 Excess of Beatty-function of 1/sin(1) over n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

Zak Seidov, Jun 13 2005

nonn

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120.

a(n) = A108120[n] - n.

A126104 Numbers n not divisible by 6 such that sigma(n) > 3n.

Original entry on oeis.org

40040, 61600, 72800, 80080, 104720, 117040, 123200, 123760, 138320, 141680, 145600, 154000, 160160, 172480, 200200, 209440, 215600, 234080, 246400, 247520, 254800, 261800, 276640, 280280, 283360, 291200, 292600, 308000

Offset: 1

Ant King, Mar 07 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A047253 and A068403.

Select[Range[310000], Mod[#, 6] > 1 && DivisorSigma[1, #] > 3# &] (* Amiram Eldar, Dec 03 2019 *)

A261831 a(2n-1) = 2n-1; otherwise a(n) is the smallest even number not already present which is obtained from the existing terms by the rules of (3*n+1)-problem.

Original entry on oeis.org

1, 4, 3, 2, 5, 10, 7, 16, 9, 8, 11, 22, 13, 28, 15, 14, 17, 34, 19, 40, 21, 20, 23, 46, 25, 52, 27, 26, 29, 58, 31, 64, 33, 32, 35, 70, 37, 76, 39, 38, 41, 82, 43, 88, 45, 44, 47, 94, 49, 100, 51, 50, 53, 106, 55, 112, 57, 56, 59, 118, 61, 124, 63, 62, 65, 130, 67, 136, 69, 68, 71, 142, 73, 148, 75, 74, 77, 154, 79, 160, 81, 80

Offset: 1

Vladimir Shevelev, Sep 02 2015

nonn

By the rules of the (3*n+1)-problem, an even number can appear either by the operation 3*x+1 only when x is an odd number or by the division of a number of the form 4*k by 2.
Using induction as in the proof of the Theorem in A261728, one can prove that if n == 0(mod 6), then a(n) = 2*n-2; if n == 2(mod 6), then a(n) = 2*n; if n == 4(mod 6), then a(n) = n-2.
The sequence is a permutation of the positive integers not divisible by 6 (A047253).

			Let n=28. Since 28 is of the form 6*k+4 with k=4, then a(28) = 6*4+2 = 26.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A109732, A261690, A261728.

a[n_] := If[OddQ[n], n, Switch[Mod[n, 6], 0, 2n-2, 2, 2n, 4, n-2]]; Array[a, 81] (* Jean-François Alcover, Sep 02 2015, from given formula *)
LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{1,4,3,2,5,10,7,16,9,8,11,22},90] (* Harvey P. Dale, Feb 24 2025 *)

a(2*n-1) = 2*n-1, for n>=1.
a(6*k) = 12*k-2, a(6*k+2) = 12*k+4 and a(6*k+4) = 6*k+2, for k>=0.
O.g.f.:(1+x (4+x (3+x (2+x (5+x (10+x (5+x (8+x (3+x (4+x (1+2 x)))))))))))/(-1+x^6)^2.

A367018 Composite squarefree k that are not divisible by 6.

Original entry on oeis.org

10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 70, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 105, 106, 110, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161, 165, 166, 170, 177, 178, 182

Offset: 1

Michael De Vlieger, Jan 15 2024

The asymptotic density of this sequence is 1/(2*Pi^2). - Amiram Eldar, Jan 20 2024

			70 is in this sequence since it is composite and squarefree but not divisible by 6. It does not appear in A006881 since it is the product of 3 primes.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A047253, A120944, A367455.

Select[Range[180], And[SquareFreeQ[#], CompositeQ[#], Mod[#, 6] != 0] &]

Intersection of A047253 and A120944 = { k : Omega(k) > omega(k) = 1, k mod 6 != 0 }.

cp's OEIS Frontend

A122841 Greatest k such that 6^k divides n.

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A097325 Period 6: repeat [0, 1, 1, 1, 1, 1].

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A248910 Numbers with no zeros in base-6 representation.

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A108120 Floor[n*1/Sin[1]], or Beatty sequence for 1/sin(1).

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A367455 Numbers not divisible by 6 that are neither squarefree nor prime powers.

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A232100 a(n) = the position of A232099(n) in A055926.

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A108611 Excess of Beatty-function of 1/sin(1) over n.

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A126104 Numbers n not divisible by 6 such that sigma(n) > 3n.

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A261831 a(2n-1) = 2n-1; otherwise a(n) is the smallest even number not already present which is obtained from the existing terms by the rules of (3*n+1)-problem.