A081604 -id:A081604 - cp's OEIS frontend

A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

2, 23, 71, 26, 77, 233, 239, 719, 701, 647, 725, 2159, 2177, 2158, 2157, 5822, 5741, 6551, 6476, 6532, 6531, 18944, 19436, 19655, 19601, 19673, 19653, 58310, 58309, 58316, 58967, 59021, 58964, 157211, 157217, 174950, 176408, 176407, 176903, 177065, 177064, 471653, 511511

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 3^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-3 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 2-digit base-3 number has at least 1 nonprime substring. Hence, each 2(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 3^(2n+1) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 2, since 2 = 2_3 (base-3) is the greatest number with zero nonprime substrings in base-3 representation.
a(1) = 23 = 212_3 has 1 substring in base-3 representation (= 1). All the other base-3 substrings (2, 2, 21, 12, 212) are prime substrings. 23 is the greatest number with 1 nonprime substring.
a(2) = 71 = 2122_3 has 10 substrings in base-3 representation (1, 2, 2, 2, 12, 21, 22, 122, 212, 2122), exactly 2 of them are nonprime substrings (1 and 22_3=8), and there is no greater number with 2 nonprime substrings in base-3 representation.
a(3) = 26 = 222_3 has 6 substrings in base-3 representation, only 3 of them are prime substrings (2, 2, 2) which implies that exactly 3 substrings must be nonprime, and there is no greater number with 3 nonprime substrings in base-3 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..250

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217103(n).
a(n) >= A217303(A000217(A081604(a(n)))-n).
Example: a(12)=2177=2222122_3, A000217(A081604(2177))=28, hence a(12)>=A217303(28-12)=1934.
a(n) <= 3^min(n + 2, 5*floor((n+4)/5)).
a(n) <= 3^(n + 2).
a(n) <= 3^min((n + 11)/3, 11*floor((n+32)/33)).
a(n) <= 3^((1/3)*(n + 11)).
With m := floor(log_3(a(n))) + 1:
a(n+m+1) >= 3*a(n), if a(n)!=1 (mod 3).
a(n+m) >= 3*a(n), if a(n)=1 (mod 3).

A262438 Number of digits of hexadecimal representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Reinhard Zumkeller, Sep 22 2015

Length of n-th row in A262437.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A262437, A055642, A070939, A081604.

Haskell
```
a262438 = length . a262437_row
```

A262438 := proc(n)
    if n =0 then
        1;
    else
        1+floor(log[16](n)) ;
    end if;
end proc: # R. J. Mathar, Dec 14 2015

a(n) = if (n, #digits(n, 16), 1); \\ Michel Marcus, Dec 14 2015

A370932 For any number n >= 0 with ternary expansion Sum_{i >= 0} t_i * 3^i, a(n) = Sum_{i >= 0} ((Sum_{j >= 0} (-1)^j * t_{i+j}) mod 3) * 3^i.

Original entry on oeis.org

0, 1, 2, 5, 3, 4, 7, 8, 6, 16, 17, 15, 9, 10, 11, 14, 12, 13, 23, 21, 22, 25, 26, 24, 18, 19, 20, 50, 48, 49, 52, 53, 51, 45, 46, 47, 27, 28, 29, 32, 30, 31, 34, 35, 33, 43, 44, 42, 36, 37, 38, 41, 39, 40, 70, 71, 69, 63, 64, 65, 68, 66, 67, 77, 75, 76, 79, 80

Offset: 0

Rémy Sigrist, Mar 06 2024

In other words, the k-th ternary digit of a(n) is congruent (modulo 3) to the alternate sum of the digits to the left of (and including) the k-th ternary digit of n.

This sequence is a permutation of the nonnegative integers with inverse A071770 that preserves the number of ternary digits (A081604) and the leading ternary digit (A122586).

			For n = 42: the ternary expansion of 42 is "1120"; also:
     + 1             = 1 (mod 3)
     - 1 + 1         = 0 (mod 3)
     + 1 - 1 + 2     = 2 (mod 3)
     - 1 + 1 - 2 + 0 = 1 (mod 3)
- so the ternary expansion of a(42) is "1021", and a(42) = 34.

Cf. A006068 (base-2 analog), A081604, A105529, A122586, A071770 (inverse).

a(n, base = 3) = { my (d = digits(n, base), s = 0); for (i = 1, #d, d[i] = (s = d[i]-s) % base;); fromdigits(d, base); }

from itertools import accumulate
from sympy.ntheory import digits
def A370932(n):
    t = accumulate(((-j if i&1 else j) for i, j in enumerate(digits(n,3)[1:])),func=lambda x,y: (x+y)%3)
    return int(''.join(str(-d%3 if i&1 else d) for i,d in enumerate(t)),3) # Chai Wah Wu, Mar 08 2024

A081604(a(n)) = A081604(n).

A122586(a(n)) = A122586(n).

A020911 Number of digits in the base 3 representation of n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 23, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30

Offset: 1

Clark Kimberling

Cf. A000045 (Fibonacci numbers), A081604 (base-3 digits of n).

[#Intseq(Fibonacci(n),3):n in [1..70]]; // Marius A. Burtea, Jan 02 2020

IntegerLength[Fibonacci[Range[50]],3] (* Harvey P. Dale, Aug 13 2018 *)

a(n)={1 + logint(fibonacci(n), 3)} \\ Andrew Howroyd, Jan 01 2020

a(n) = A081604(A000045(n)). - Andrew Howroyd, Jan 01 2020

Offset corrected and terms a(50) and beyond from Andrew Howroyd, Jan 01 2020

A037862 a(n)=(number of digits <=1)-(number of digits >1) in base 3 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Cf. A081604, A081603.

A037862 := proc(n)
    a := 0 ;
    dgs := convert(n,base,3);
    for i from 1 to nops(dgs) do
        if op(i,dgs)<=1 then
            a := a+1 ;
        else
            a := a-1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

b3d[n_]:=Module[{idn=IntegerDigits[n,3]},Count[idn,?(#<2&)]-Count[idn, ?(#>1&)]]; Array[b3d,90] (* Harvey P. Dale, Nov 08 2011 *)

a(n) = A081604(n)-2*A081603(n). - R. J. Mathar, Jan 27 2025

A305878 For any number n >= 0: apply the map 0 -> "0", 1 -> "01", 2 -> "011" to the ternary representation of n and interpret the result as a binary string.

Original entry on oeis.org

0, 1, 3, 2, 5, 11, 6, 13, 27, 4, 9, 19, 10, 21, 43, 22, 45, 91, 12, 25, 51, 26, 53, 107, 54, 109, 219, 8, 17, 35, 18, 37, 75, 38, 77, 155, 20, 41, 83, 42, 85, 171, 86, 173, 347, 44, 89, 179, 90, 181, 363, 182, 365, 731, 24, 49, 99, 50, 101, 203, 102, 205, 411

Offset: 0

Rémy Sigrist, Jun 13 2018

This sequence is a ternary analog of A048678.

This sequence is a permutation of A003726.

			The first terms, alongside the ternary representation of n and the binary representation of a(n), are:
  n   a(n)  tern(n)  bin(a(n))
  --  ----  -------  ---------
   0     0        0          0
   1     1        1          1
   2     3        2         11
   3     2       10         10
   4     5       11        101
   5    11       12       1011
   6     6       20        110
   7    13       21       1101
   8    27       22      11011
   9     4      100        100
  10     9      101       1001
  11    19      102      10011
  12    10      110       1010

Rémy Sigrist, Colored logarithmic scatterplot of the first 3^10 terms (where the color is function of A053735(n) + A081604(n))

Cf. A000120, A003726, A048678, A053735, A081604.

a(n) = if (n==0, 0, my (d=n%3); a(n\3) * 2^(d+1) + (2^d-1))

a(0) = 0.
a(3*n) = 2*a(n).
a(3*n + 1) = 4*a(n) + 1.
a(3*n + 2) = 8*a(n) + 3.
A000120(a(n)) = A053735(n).

A318488 a(0) = 0, a(n) = -5*a(n/3) if n is divisible by 3, otherwise a(n) = n + a(n-1).

Original entry on oeis.org

0, 1, 3, -5, -1, 4, -15, -8, 0, 25, 35, 46, 5, 18, 32, -20, -4, 13, 75, 94, 114, 40, 62, 85, 0, 25, 51, -125, -97, -68, -175, -144, -112, -230, -196, -161, -25, 12, 50, -90, -50, -9, -160, -117, -73, 100, 146, 193, 20, 69, 119, -65, -13, 40, -375, -320, -264, -470, -412, -353, -570, -509, -447, -200, -136, -71, -310

Offset: 0

Altug Alkan, Aug 27 2018

From a generalization of A318303 (compare the scatterplots in order to observe connection). In this case, A000244 is determinative for the boundaries of self-similar block structures of this sequence, i.e., n = 3^9 - 1 is a corresponding endpoint.

Altug Alkan, Table of n, a(n) for n = 0..19682
Rémy Sigrist, Colored scatterplot of a(n) for n = 0..3^10-1 (where the color is function of floor(n / 3^(A081604(n)-5)))

Cf. A081604, A318303.

[0] cat [n eq 1 select 1 else n mod 3 eq 0 select -5*Self(n div 3) else Self(n-1)+n: n in [1..70]]; // Vincenzo Librandi, Aug 28 2018

a[0]=0; a[n_] := a[n] = If[Mod[n, 3] == 0, -5 a[n/3], n + a[n - 1]]; Array[a, 70, 0] (* Giovanni Resta, Aug 27 2018 *)

a(n)=if(n==0, 0, if(n%3, n+a(n-1), -5*a(n/3)));

A280724 Expansion of 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(3^k).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229, 233, 237, 241, 245

Offset: 0

Ilya Gutkovskiy, Jan 07 2017

Sums of lengths of ternary numbers (A007089).

			-----------------------
n  base 3 length  a(n)
-----------------------
0 |  0   |  1   |  1
1 |  1   |  1   |  2
2 |  2   |  1   |  3
3 |  10  |  2   |  5
4 |  11  |  2   |  7
5 |  12  |  2   |  9
6 |  20  |  2   |  11
7 |  21  |  2   |  13
8 |  22  |  2   |  15
9 |  100 |  3   |  18
-----------------------

Eric Weisstein's World of Mathematics, Ternary

Cf. A007089, A062153, A081604, A083652, A117804.

CoefficientList[Series[1/(1 - x) + (1/(1 - x)^2) Sum[x^3^k, {k, 0, 15}], {x, 0, 70}], x]
Table[1 + Sum[Floor[Log[3, k]] + 1, {k, 1, n}], {n, 0, 70}]

G.f.: 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(3^k).

a(n) = 1 + Sum_{k=1..n} floor(log_3(k)) + 1.

cp's OEIS Frontend

A217113 Greatest number (in decimal representation) with n nonprime substrings in base-3 representation (substrings with leading zeros are considered to be nonprime).

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A262438 Number of digits of hexadecimal representation of n.

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A370932 For any number n >= 0 with ternary expansion Sum_{i >= 0} t_i * 3^i, a(n) = Sum_{i >= 0} ((Sum_{j >= 0} (-1)^j * t_{i+j}) mod 3) * 3^i.

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A020911 Number of digits in the base 3 representation of n-th Fibonacci number.

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A037862 a(n)=(number of digits <=1)-(number of digits >1) in base 3 representation of n.

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A305878 For any number n >= 0: apply the map 0 -> "0", 1 -> "01", 2 -> "011" to the ternary representation of n and interpret the result as a binary string.

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A318488 a(0) = 0, a(n) = -5*a(n/3) if n is divisible by 3, otherwise a(n) = n + a(n-1).

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A280724 Expansion of 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(3^k).

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