A051064 -id:A051064 - cp's OEIS frontend

A335933 A fractal function, related to ruler functions. a(1) = 0; otherwise for m >= 0, a(3m) = 1, a(3m-1) = a(2m-1) + sign(a(2m-1)), a(3m+1) = a(2m+1) + sign(a(2m+1)).

1, 0, 0, 1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 6, 6, 1, 4, 4, 1, 2, 2, 1, 3, 3, 1, 7, 7, 1, 2, 2, 1, 5, 5, 1, 3, 3, 1, 2, 2, 1, 4, 4, 1, 8, 8, 1, 2, 2, 1, 3, 3, 1, 6, 6, 1, 2, 2, 1, 4, 4, 1, 3, 3, 1, 2, 2, 1, 5, 5, 1

Offset: 0

Peter Munn, Jun 30 2020

We choose a form for the definition that shows clearly its relationship to A307744.
The odd bisection is essentially A087088.
If we add a(-1) = 0 to the definition and allow negative m (and therefore n), we get a symmetric function, that is a(n) = a(-n).
For k >= 1 numbers 1..k occur with the same periodic and mirror symmetries as in A307744 and in ruler function A051064. In A051064, k occurs 3 times more frequently than k+1. Here, and in A307744, k occurs 3/2 times more frequently than k+1, precisely 2^(k-1) times in every 3^k terms.

Sequences with similar definitions: A205593, A307744.
A051064 has matching symmetries.
Odd bisection: A087088.

a(n) = if (n==1, 0, if ((n%3) == 0, 1, if ((n%3)==1, my(k=(n-1)/3, aa = a(2*k+1)); aa+sign(aa),  my(k=(n+1)/3, aa = a(2*k-1)); aa+sign(aa)))); \\ Michel Marcus, Jul 03 2020

A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.

Original entry on oeis.org

1, 2, 6, 4, 5, 12, 7, 8, 27, 10, 11, 24, 13, 14, 30, 16, 17, 54, 19, 20, 42, 22, 23, 48, 25, 26, 108, 28, 29, 60, 31, 32, 66, 34, 35, 108, 37, 38, 78, 40, 41, 84, 43, 44, 135, 46, 47, 96, 49, 50, 102, 52, 53, 216, 55, 56, 114, 58, 59, 120, 61, 62, 189, 64, 65, 132, 67, 68, 138, 70, 71, 216, 73, 74

Offset: 1

Seiichi Manyama, Jul 14 2023

Cf. A007949, A051064, A327625.

Cf. A091512, A364223, A364224.

a[n_] := n * (IntegerExponent[n, 3] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

PARI
```
a(n) = n*(valuation(n, 3)+1);
```

a(n) = n * A051064(n).
If n == 0 (mod 3), a(n) = n + 3 * a(n/3) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(3^e) = (e+1)*3^e and a(p^e) = p*e if p != 3.
Dirichlet g.f.: (3^s/(3^s-3)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4)*n^2. (End)

A168570 Exponent of 3 in 2^n - 1.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Martins Opmanis, Nov 30 2009

nonn

Records: a(A025192(n)) = n and a(k) < n for k < A025192(n). [Joerg Arndt, Apr 07 2014]

			For n=6, 2^6 - 1 = 63. Greatest divisor of 63 which is a power of 3 is 9 (3^2).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A051064 (without the zeros).

a:= n-> padic[ordp](2^n-1, 3):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Table[IntegerExponent[2^n - 1, 3], {n, 100}] (* T. D. Noe, Apr 13 2014 *)

vector(100,n,valuation(2^n-1,3)) /* Joerg Arndt, Jun 13 2011 */

A332202 Largest k >= 0 such that 3^k divides 2^(2^n-1) + 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1

Offset: 0

M. F. Hasler, Mar 05 2020

nonn

Behaves like a mixture of 2-adic and 3-adic ruler function, cf. formula.

			a(0) = 0 since 2^(2^0-1) + 1 = 2^0 + 1 = 2 is not divisible by 3.
a(1) = 1 since 2^(2^1-1) + 1 = 2^1 + 1 = 3 is divisible just once by 3.
a(2) = 2 since 2^(2^2-1) + 1 = 2^3 + 1 = 9 is divisible by 3^2.
a(3) = 1 since 2^(2^4-1) + 1 = 2^15 + 1 = 32769 is divisible only once by 3.

Cf. A007949, A051064, A001511 (2-adic ruler)

apply( {A332202(n)=if(bittest(n,0), 1, n, valuation(n\2,3)+2)}, [0..99])

For all n > 0, a(2n-1) = 1; a(2n) = 2 + A007949(n) = 1 + A051064(n).

A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 6, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

Hugo Pfoertner, Oct 28 2023

1

Cf. A002283, A003020, A007949, A051064, A057951, A102347, A366921.

a[n_]:=IntegerExponent[10^n-1,3]; Array[a,90] (* Stefano Spezia, Oct 28 2023 *)

PARI
```
a366922(n) = valuation(10^n-1,3)
```

def A366922(n):
    c = 0
    a, b = divmod(10**n-1, 3)
    while b == 0:
        a, b = divmod(a, 3)
        c += 1
    return c # Chai Wah Wu, Oct 29 2023

a(n) = A007949(10^n - 1).

a(n) = A007949(n) + 2 = A051064(n) + 1.

A373438 Expansion of Sum_{k>=1} k * x^(3^(k-1)) / (1 - x^(3^(k-1))).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 15, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3, 1, 1, 6, 1, 1, 3, 1, 1, 3

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

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

Cf. A000217, A051064, A080278, A115364.

nmax = 105; CoefficientList[Series[Sum[k x^(3^(k - 1))/(1 - x^(3^(k - 1))), {k, 1, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
Table[Binomial[IntegerExponent[3 n, 3] + 1, 2], {n, 1, 105}]

a(n) = {my(e = valuation(n, 3)); (e+1)*(e+2)/2;} \\ Amiram Eldar, Jun 27 2024

a(n) = A000217(A051064(n)).
From Vaclav Kotesovec, Jun 25 2024: (Start)
Dirichlet g.f.: zeta(s) * (3^s/(3^s-1))^2.
Sum_{k=1..n} a(k) ~ 9*n/4 - log(n)*(log(n) + 2*log(6*Pi))/(4*log(3)^2). (End)
Multiplicative with a(p^e) = (e+1)*(e+2)/2 if p = 3 and 1 if p != 3. - Amiram Eldar, Jun 27 2024

A382487 The number of divisors of n whose largest prime factor is 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 4, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 5, 0, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 3, 0, 0, 1

Offset: 1

Amiram Eldar, Mar 29 2025

The number of 3-smooth divisors of n that are not powers of 2.

The number of terms of A065119 that divide n.

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

Cf. A001511, A001651, A065119, A072078, A007949, A051064, A169611, A301461, A306771.

a[n_] := (IntegerExponent[n, 2] + 1) * IntegerExponent[n, 3]; Array[a, 100]

a(n) = (valuation(n, 2) + 1) * valuation(n, 3);

a(n) = A072078(n) - A001511(n).
a(n) = A001511(n) * A007949(n).
a(n) = 0 if and only if n is in A001651.
a(n) = 1 if and only if n is in A306771.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.
In general, the asymptotic mean of the number prime(k+1)-smooth divisors of n that are not prime(k)-smooth, for k >= 1, is (1/(prime(k+1)-1)) * Product_{i=1..k} (prime(i)/(prime(i)-1)).
Dirichlet g.f.: (zeta(s)/(1-1/2^s))*(1/(1-1/3^s) - 1).

A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 81, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3, 243, 3, 3, 9, 3, 3, 9, 3, 3, 27, 3, 3, 9, 3, 3, 9, 3, 3

Offset: 1

Benoit Cloitre, Sep 04 2002

nonn

If m == 1, 2 or 3 (mod 4) then F(m) is not divisible by 3.

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

Cf. A000045, A000203, A038500, A033888, A051064, A074724, A088838.

Table[3^IntegerExponent[Fibonacci[4n],3],{n,100}] (* Harvey P. Dale, Jun 03 2012 *)

a(n) = 3^valuation(fibonacci(4*n), 3); \\ Michel Marcus, May 13 2022

If k == 1 or 2 (mod 3) then a(3^m*k) = 3^(m+1) for m>=0.
a(n) = A038500(A033888(n)). - Amiram Eldar, May 13 2022
a(n) = 3^A051064(n) (conjectured). - Michel Marcus, May 17 2022
Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - Peter Bala, Jun 10 2022

A122005 Triangle read by rows: n-th row starts with n and continues with 1/3 the previous value as long as that is an integer.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 6, 2, 7, 8, 9, 3, 1, 10, 11, 12, 4, 13, 14, 15, 5, 16, 17, 18, 6, 2, 19, 20, 21, 7, 22, 23, 24, 8, 25, 26, 27, 9, 3, 1, 28, 29, 30, 10, 31, 32, 33, 11, 34, 35, 36, 12, 4, 37, 38, 39, 13, 40, 41, 42, 14, 43, 44, 45, 15, 5, 46, 47, 48, 12, 4, 49, 50, 51, 17

Offset: 1

Jonathan Vos Post, Oct 14 2006

A fractal sequence, which is to 3 as A123390 is to 2. Row lengths are A051064 3^a(n) exactly divides 3*n. Or, 3-adic valuation of 3*n.

			Triangle starts:
1;
2;
3, 1;
4;
5;
6, 2;
7;
8;
9, 3, 1;
10;
11;
12, 4;
13;
14;
15, 5;
16;

Cf. A051064, A123390.

Flatten[Function[n,NestWhile[Append[#, Last[#]/3] &, {n}, Last[#]/3 == Floor[Last[#]/3] &]][#] & /@ Range[50]] (* Birkas Gyorgy, Apr 14 2011 *)

a(1) = 1, for n > 1, if 3|a(n-1) then a(n) = a(n-1)/3, otherwise a(n) = (max_{k

A284413 Exponent of 3 in 2^n + 1.

Original entry on oeis.org

1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2

Offset: 1

Jon E. Schoenfield, Mar 26 2017

Records: a(3^(n-1)) = n and a(k) < n for k < 3^(n-1).

Multiplicative because A051064 is. - Andrew Howroyd, Jul 28 2018

			a(27) = 4 because 2^27 + 1 = 134217729 = 3^4 * 19 * 87211.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Indranil Ghosh)

Cf. A051064, A168570 (exponent of 3 in 2^n - 1).

[IsEven(n) select 0 else Factorization(3*n)[1][2]: n in [1..87]];

Table[If[OddQ[n], IntegerExponent[3n, 3], 0], {n, 100}] (* Indranil Ghosh, Mar 27 2017 *)

a(n) = if(n%2, if(n<1, 0, 1 + valuation(n, 3)), 0); \\ Indranil Ghosh, Mar 27 2017

a(n) = A051064(n) if n is odd, 0 otherwise.
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Sep 11 2020
From Amiram Eldar, Dec 27 2022: (Start)
Multiplicative with a(2^e) = 0, a(3^e) = e+1, and a(p^e) = 1 if p >= 5.
Dirichlet g.f.: zeta(s)*(1-1/2^s)/(1-1/3^s). (End)

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A335933 A fractal function, related to ruler functions. a(1) = 0; otherwise for m >= 0, a(3m) = 1, a(3m-1) = a(2m-1) + sign(a(2m-1)), a(3m+1) = a(2m+1) + sign(a(2m+1)).

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A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.

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A168570 Exponent of 3 in 2^n - 1.

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A332202 Largest k >= 0 such that 3^k divides 2^(2^n-1) + 1.

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A366922 a(n) is the exponent of 3 in the prime factorization of 10^n - 1.

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A373438 Expansion of Sum_{k>=1} k * x^(3^(k-1)) / (1 - x^(3^(k-1))).

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A382487 The number of divisors of n whose largest prime factor is 3.

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A074724 Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.

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