A069352 -id:A069352 - cp's OEIS frontend

A086413 Duplicate of A069352.

0, 1, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 5, 4, 5, 4, 6, 5, 4, 6, 5, 7, 6, 5, 7, 6, 5, 8, 7, 6, 8, 7, 6, 9, 8

Offset: 1

dead

A022328 Exponent of 2 (value of i) in n-th number of form 2^i*3^j, i >= 0, j >= 0 (see A003586).

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

Offset: 1

Clark Kimberling

nonn

From Clark Kimberling, Mar 18 2015 and May 21 2015: (Start)
This is the signature sequence of log(3)/log(2) and is a fractal sequence; e.g., if the first occurrence of each n is removed, the resulting sequence is the original sequence.
Moreover, if the sequence is partitioned into segments starting with 0 as follows:
  0,1
  0,2,1,3
  0,2,4,1,3
  0,5,2,4,1,6,3,
and so on, then deleting the greatest number in each segment leaves
  0
  0,2,1
  0,2,1,3
  0,5,2,4,1,3,
and so on, which, concatenated to (0,0,2,1,0,2,1,3,0,5,2,4,1,3,...), is another fractal sequence, in today's usual meaning of that term.  When introduced in 1995, one of the defining properties of a fractal sequence was, essentially, that before each n appears, every k < n must have already appeared; this requirement ensures that the sequence yields a dispersion; e.g., A114577 yields A114537.  However, the usual meaning of "fractal sequence" nowadays is simply "a sequence that contains itself as a proper subsequence".  It is proposed here that the original version be renamed "strongly fractal".  Thus, the operations called upper trimming and lower trimming (e.g., A084531, A167237), when applied to strongly fractal sequences, yield strongly fractal sequences.  The operation introduced here, which can be called "segment-upper trimming", carries fractal sequences to fractal sequences, but not strongly fractal to strongly fractal.
Associated with the signature sequence S of each positive irrational number is an interspersion (or equivalently, a dispersion), in which row n >= 0 consists of the positions of n in S. The interspersion associated with the signature sequence of log(3)/log(2) is A255975.
(End)
Comment from Allan C. Wechsler, May 26 2024 (Start):
More generally, the "signature sequence" of an irrational number H can be defined as follows. Consider all the numbers of the form a + bH, where a and b are positive integers, and sort them into increasing order (there are no cluster points or other obstacles). The sequence of a-values is then the *signature sequence of H.
If the coefficients a and b are allowed to be 0, you get the same sequence but with all the entries decremented by 1.
(End)
a(n) = A069352(n) - A022329(n). - Reinhard Zumkeller, May 16 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Franklin T. Adams-Watters)
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.

Cf. A003586, A033485, A022329, A191475. - Franklin T. Adams-Watters, Mar 19 2009

Cf. A069352.

import Data.Set (singleton, deleteFindMin, insert)
a022328 n = a022328_list !! (n-1)
(a022328_list, a022329_list) = unzip $ f $ singleton (1, (0, 0)) where
   f s = (i, j) :
         f (insert (2 * y, (i + 1, j)) $ insert (3 * y, (i, j + 1)) s')
         where ((y, (i, j)), s') = deleteFindMin s
-- Reinhard Zumkeller, Nov 19 2015, May 16 2015

t = Sort[Flatten[Table[2^i 3^j, {i, 0, 200}, {j, 0, 200}]]];
Table[IntegerExponent[t[[n]], 2], {n, 1, 200}]  (* A022338 *)
(* Clark Kimberling, Mar 18 2015 *)

from sympy import integer_log
def A022328(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return (~(m:=bisection(f,n,n))&m-1).bit_length() # Chai Wah Wu, Sep 15 2024

a(n) = A191475(n) - 1. - Franklin T. Adams-Watters, Mar 19 2009 [Corrected by N. J. A. Sloane, May 26 2024]

A003586(n) = 2^a(n)*3^A022329(n). - N. J. A. Sloane, Mar 19 2009

A022329 Exponent of 3 (value of j) in n-th number of form 2^i*3^j (see A003586).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 3, 0, 2, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 5, 0, 2, 4, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4, 6, 1, 3, 5, 0, 7, 2, 4, 6, 1, 3, 5, 0, 7, 2, 4, 6, 1, 8, 3, 5, 0, 7, 2, 4, 6, 1, 8, 3, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 10, 5, 0, 7, 2, 9, 4, 6, 1, 8, 3, 10, 5, 0, 7

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Franklin T. Adams-Watters)

Cf. A003586, A022328, A191476. - Franklin T. Adams-Watters, Mar 19 2009

Cf. A069352.

a022329 n = a022329_list !! (n-1)
-- Where a022329_list is defined in A022328.
-- Reinhard Zumkeller, Nov 19 2015, May 16 2015

IntegerExponent[Select[Range[10^5], # == 2^IntegerExponent[#, 2] * 3^IntegerExponent[#, 3] &], 3] (* Amiram Eldar, Apr 15 2024 *)

from sympy import integer_log
def A022329(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return integer_log((m:=bisection(f,n,n))>>(~m&m-1).bit_length(),3)[0] # Chai Wah Wu, Sep 15 2024

a(n) = A069352(n) - A022328(n). - Reinhard Zumkeller, May 16 2015
A003586(n) = 2^A022328(n) * 3^a(n). - N. J. A. Sloane, Mar 19 2009
a(n) = A191476(n) - 1. - Franklin T. Adams-Watters, Mar 19 2009

Edited by N. J. A. Sloane, May 26 2024

A112759 Total number of prime factors of n-th 5-smooth number.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 2, 4, 3, 3, 4, 2, 3, 3, 5, 4, 4, 3, 5, 3, 4, 4, 6, 5, 3, 5, 4, 4, 6, 4, 5, 5, 3, 7, 4, 6, 4, 6, 5, 5, 7, 5, 6, 4, 6, 5, 4, 8, 5, 7, 5, 7, 6, 6, 4, 8, 6, 5, 7, 5, 7, 6, 5, 9, 6, 8, 6, 4, 8, 7, 5, 7, 6, 5, 9, 7, 6, 8, 6, 8, 7, 6, 10, 7, 5, 9, 7, 6, 5, 9, 8, 6, 8, 7, 6, 10, 8, 7, 9, 7

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A001222, A051037, A069352, A112754, A112758, A112760, A112761, A112762.

PrimeOmega @ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A001222(A051037(n));

a(n) = A112760(n) + A112761(n) + A112762(n).

A086415 Maximal exponent in prime factorization of 3-smooth numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

a(n) = A051903(A003586(n));

A086414(n) <= a(n) <= A069352(n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003586, A051903.

Cf. A086414, A069352.

N:= 10^20: # to include all 3-smooth numbers <= N
S:= [seq(seq([2^i*3^j,max(i,j)], j=0..floor(log[3](N/2^i))),i=0..floor(log[2](N)))]:
map(p -> p[2], sort(S,(a,b) -> a[1]Robert Israel, Aug 10 2014

M = 10^5; (* M = 10^5 gives 101 terms *)
S = Flatten[Table[Table[{2^i*3^j, Max[i, j]}, {j, 0, Floor[Log[3, M/2^i]]}], {i, 0, Floor[Log[2, M]]}], 1] // Sort;
S[[All, 2]] (* Jean-François Alcover, Mar 03 2019, after Robert Israel *)

A036667 Numbers of the form 2^i*3^j, i+j even.

Original entry on oeis.org

1, 4, 6, 9, 16, 24, 36, 54, 64, 81, 96, 144, 216, 256, 324, 384, 486, 576, 729, 864, 1024, 1296, 1536, 1944, 2304, 2916, 3456, 4096, 4374, 5184, 6144, 6561, 7776, 9216, 11664, 13824, 16384, 17496, 20736, 24576, 26244, 31104, 36864, 39366

Offset: 1

N. J. A. Sloane

nonn

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

Complement of A257999 with respect to A003586.
Intersection of A028260 and A003586.
Cf. A025620 (subsequence), A069352, A022328, A022329.

a036667 n = a036667_list !! (n-1)
a036667_list = filter (even . flip mod 2 . a001222) a003586_list
-- Reinhard Zumkeller, May 16 2015

max = 40000;
Reap[Do[k = 2^i 3^j; If[k <= max && EvenQ[i+j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}]][[2, 1]] // Union (* Jean-François Alcover, Aug 04 2018 *)

from sympy import integer_log
def A036667(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length()+(i&1^1)>>1 for i in range(integer_log(x, 3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

A069352(a(n)) mod 2 = 0. - Reinhard Zumkeller, May 16 2015

Sum_{n>=1} 1/a(n) = 7/4. - Amiram Eldar, Feb 18 2021

Offset corrected by Reinhard Zumkeller, May 16 2015

A112754 Total number of prime factors of n-th number of the form 3^i*5^j.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A001222, A003593, A022336, A022337, A069352, A112753, A112759.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; PrimeOmega[Union[s]] (* Amiram Eldar, Feb 06 2020 *)

a(n) = A001222(A003593(n)) = A022336(n) + A022337(n).

A086414 Minimal exponent in prime factorization of 3-smooth numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A003586, A051904, A069352, A086415.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; maxExp[1] = 0; maxExp[n_] := Min @@ Last /@ FactorInteger[n]; maxExp /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A051904(A003586(n));

a(n) <= A086415(n) <= A069352(n).

A086412 Number of distinct prime factors of 3-smooth numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A001221, A003586, A069352, A086414.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; PrimeNu /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A001221(A003586(n));

a(n) <= 2.

A069355 Numbers of form 2^i*3^j - (i+j) with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 12, 15, 20, 24, 27, 32, 43, 50, 58, 67, 77, 90, 103, 121, 138, 157, 185, 210, 238, 248, 281, 318, 376, 425, 480, 503, 568, 641, 723, 759, 856, 965, 1014, 1143, 1288, 1451, 1526, 1719, 1936, 2037, 2180, 2294, 2583, 2908, 3061, 3446, 3879, 4084

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Distinct values of A003586(m)-A069352(m) or of A069345(A003586(m)). - Michel Marcus, Apr 09 2018

			1 is a term because 2^0*3^0 - (0+0) = 2^1*3^0 - (1+0) = 1.
2 is a term because 2^2*3^0 - (2+0) = 2^0*3^1 - (0+1) = 2.
4 is a term because 2^1*3^1 - (1+1) = 4.

Cf. A003586, A069345, A069352, A069353, A069356, A069357, A022328, A022329.

With[{nn=20},Take[Flatten[Table[2^i 3^j-i-j,{i,0,nn},{j,0,nn}]]//Union,60]] (* Harvey P. Dale, Aug 29 2022 *)

Duplicated term 2 and incorrect formula removed by Altug Alkan, Apr 09 2018

cp's OEIS Frontend

A086413 Duplicate of A069352.

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A022328 Exponent of 2 (value of i) in n-th number of form 2^i*3^j, i >= 0, j >= 0 (see A003586).

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A022329 Exponent of 3 (value of j) in n-th number of form 2^i*3^j (see A003586).

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A112759 Total number of prime factors of n-th 5-smooth number.

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A086415 Maximal exponent in prime factorization of 3-smooth numbers.

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A036667 Numbers of the form 2^i*3^j, i+j even.

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A112754 Total number of prime factors of n-th number of the form 3^i*5^j.

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A086414 Minimal exponent in prime factorization of 3-smooth numbers.

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A086412 Number of distinct prime factors of 3-smooth numbers.

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