A003586 -id:A003586 - cp's OEIS frontend

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

A022330 Index of 3^n within sequence of numbers of form 2^i*3^j (A003586).

Original entry on oeis.org

1, 3, 7, 12, 19, 27, 37, 49, 62, 77, 93, 111, 131, 152, 175, 199, 225, 252, 281, 312, 344, 378, 413, 450, 489, 529, 571, 614, 659, 705, 753, 803, 854, 907, 961, 1017, 1075, 1134, 1195, 1257, 1321, 1386, 1453, 1522, 1592, 1664, 1737, 1812, 1889, 1967, 2047, 2128

Offset: 0

Clark Kimberling

nonn

a(1000)=793775, a(10000)=79261054, a(100000)=7924941755, a(1000000)=792482542841.

Zak Seidov, Table of n, a(n) for n = 0..10000 (terms for n = 0..1000 from Charles R Greathouse IV).
N. Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, arXiv preprint arXiv:1303.0888 [math.CO], 2013.
N. Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, J. Int. Seq. 16 (2013) #13.3.4.

Cf. A022331, A020914 (first differences).

Cf. A000244, A003586, A071521.

c[0] = 1; c[n_] := 1 + Sum[Ceiling[j*Log[2, 3]], {j, n}]; Table[c[i], {i, 0, 51}] (* Norman Carey, Jun 13 2012 *)

listsm(lim)=my(v=List(),N); for(n=0,log(lim)\log(3),N=3^n; while(N<=lim,listput(v,N);N<<=1)); v=Vec(v); vecsort(v)
list(lim)=my(v=listsm(3^floor(lim)));vector(floor(lim+1),i,setsearch(v,3^(i-1))) \\ Charles R Greathouse IV, Aug 19 2011

a(n)=sum(k=0,n, logint(3^k,2))+n+1 \\ Charles R Greathouse IV, Nov 22 2022

def A022330(n): return sum((3**i).bit_length() for i in range(n+1)) # Chai Wah Wu, Sep 16 2024

a(n) = A071521(A000244(n)); A003586(a(n)) = A000244(n). - Reinhard Zumkeller, May 09 2006
a(n) ~ kn^2 with k = log(3)/log(4) = 0.792.... More exact asymptotics? - Zak Seidov, Dec 22 2011
a(n+1) = a(n) + A020914(n+1). - Ruud H.G. van Tol, Nov 25 2022
kn^2 + kn + 1 <= a(n) <= kn^2 + (k+1)n + 1, so a(n) = kn^2 + O(n) with k = log(3)/log(4). The law of the iterated logarithm suggests that a better error term might be possible. - Charles R Greathouse IV, Nov 28 2022

A022331 Index of 2^n within sequence of numbers of form 2^i*3^j (A003586).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 48, 56, 65, 74, 84, 95, 106, 118, 130, 143, 157, 171, 186, 202, 218, 235, 253, 271, 290, 309, 329, 350, 371, 393, 416, 439, 463, 487, 512, 538, 564, 591, 619, 647, 676, 706, 736, 767, 798, 830, 863, 896, 930, 965, 1000, 1036, 1072

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)
Norman Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, arXiv preprint arXiv:1303.0888 [math.CO], 2013; Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, J. Int. Seq. 16 (2013), Article 13.3.4.

Cf. A000079, A003586, A071521, A020915 (first differences), A152747.

Cf. A022330 (index of 3^n within A003586).

c[0] = 1; c[n_] := 1 + Sum[Ceiling[j*Log[3, 2]], {j, n}]; Table[c[i], {i, 0, 60}] (* Norman Carey, Jun 13 2012 *)

a(n)=my(t=1);1+n+sum(k=1,n,logint(t*=2,3)) \\ Ruud H.G. van Tol, Nov 25 2022

from sympy import integer_log
def A022331(n):
    m = 1<Chai Wah Wu, Sep 16 2024

a(n) = A071521(A000079(n)); A003586(a(n)) = A000079(n). - Reinhard Zumkeller, May 09 2006

a(n) ~ c * n^2, where c = log(2)/(2*log(3)) (A152747). - Amiram Eldar, Apr 07 2023

A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 52, 1088, 84, 308, 1092, 116, 5184, 820, 1108, 372, 5188, 2868, 1140, 13376, 884, 5204, 17204, 1396, 13380, 2932, 5236, 275520, 19252, 1908, 13396, 17268, 5492, 275524, 84788, 3956, 13428, 1324096, 19316, 6004, 275540, 215860, 18292, 13684

Offset: 1

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   64: {{1,2,3}}
   20: {{1,2},{1,3}}
   68: {{1,2},{1,2,3}}
   52: {{1,2},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

With distinctness we have A367910, sorted A367911, firsts of A367905.
For multisets we have A367913, sorted A367915, firsts of A367912.
Positions of first appearances in A368109.
The sorted version is A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dd=Select[Range[nn],Max@@First/@FactorInteger[#]<=3&];
qq=Table[Length[Tuples[bpe/@bpe[n]]],{n,nn}];
kk=Select[Range[Length[dd]],SubsetQ[qq,Take[dd,#]]&]
Table[Position[qq,dd[[n]]][[1,1]],{n,kk}]

A205506 Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

Original entry on oeis.org

2, 2, 6, 2, 3, 5, 7, 3, 4, 3, 6, 93, 2, 88, 5, 33, 5, 196, 15, 106, 174, 196, 14, 342, 207, 28, 372, 14, 47, 25, 569, 646, 141, 129, 278, 5, 421, 224, 629, 26, 424, 1081, 688, 246, 736, 4392, 124, 484, 759, 791, 4401, 863, 2854, 410, 1044, 22, 848, 1402, 2006

Offset: 1

Lei Zhou, Feb 01 2012

1 - m^k + m^(2*k) equals Phi(6*k,m) when k=2^p*3^q, p>=0, q>=0, which may be prime numbers for certain positive integer m>1.
The Mathematica program given here generates the first 33 terms.  Further terms were generated by OpenPFGW.
a(62)=7426, while A003586(62)=3^8=6561.

			n=1, A003586(1)=1, when m=2, 1-2^1+2^2=3 is prime, so a(1)=2;
n=2, A003586(2)=2, when m=2, 1-2^2+2^4=13 is prime, so a(2)=2;
...
n=7, A003586(7)=9, when m=7, 1-7^9+7^18=1628413557556843 is prime, so a(7)=7.

Cf. A003586, A033845, A056993, A085398, A153438.

fQ[n_] := n == 3 EulerPhi@n; a = Select[6 Range@500, fQ]/6; l =
Length[a]; Table[m = a[[j]]; i = 1;
While[i++; cp = 1 - i^m + i^(2*m); ! PrimeQ[cp]]; i, {j, 1, l}]

from itertools import count
from sympy import isprime, integer_log
def A205506(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))
    k = bisection(f,n,n)
    return next(filter(lambda m:isprime(1-m**k+m**(k<<1)),count(2))) # Chai Wah Wu, Oct 22 2024

a(n) = A085398(6*A003586(n)). - Jinyuan Wang, Jan 01 2023

a(n) is smallest positive m such that Phi(A033845(n),m) is prime. - Chai Wah Wu, Sep 16 2024

A276380 Irregular triangle where row n contains terms k of the partition of n produced by greedy algorithm such that all elements are in A003586.

Original entry on oeis.org

1, 2, 3, 4, 1, 4, 6, 1, 6, 8, 9, 1, 9, 2, 9, 12, 1, 12, 2, 12, 3, 12, 16, 1, 16, 18, 1, 18, 2, 18, 3, 18, 4, 18, 1, 4, 18, 24, 1, 24, 2, 24, 27, 1, 27, 2, 27, 3, 27, 4, 27, 32, 1, 32, 2, 32, 3, 32, 36, 1, 36, 2, 36, 3, 36, 4, 36, 1, 4, 36, 6, 36, 1, 6, 36, 8, 36, 9, 36, 1, 9, 36, 2, 9, 36, 48, 1, 48

Offset: 1

Michael De Vlieger, Sep 25 2016

This sequence uses a greedy algorithm f(x) to find the largest number k <= n such that k is in A003586. The function is recursively applied to the result until it reaches 1. This is the algorithm described in the reference p. 36. This sequence presents the terms in order from least to greatest term.
The reference suggests the greedy algorithm is one way to render n in a "dual-base number system", essentially base (2,3) with bases 2 and 3 arranged orthogonally to produce a matrix of places with values that are the tensor product of prime power ranges of 2 and 3. Place values are signified by 0 or 1. Thus we can boil down the matrix to simply list the values of places harboring digit 1.
Row n = n for n that are in A003586.
The reference defines a "canonic" representation of n on page 33 as having the lowest number of terms. The greedy algorithm does not always render the canonic representation. a(41) = {1,4,36}, but {9,32} is the shortest possible partition of 41 such that all terms are in A003586.
The terms in row n differ from the canonic terms at n = 41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, 185, 209, 215, 226, 236, 239, 248... (i.e., A277071).

			Triangle begins:
1
2
3
4
1,4
6
1,6
8
9
1,9
2,9
12
1,12
2,12
3,12
16
1,16
18
1,18
2,18
3,18
4,18
1,4,18
...

V. Dimitrov, G. Jullien, and R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.

Michael De Vlieger, Table of n, a(n) for n = 1..11006 (Rows 1 <= n <= 3600)

Cf. A003586, A237442 (least number of 3-smooth numbers that add up to n), A277070 (row lengths), A277071, A347860, A348599.

Table[Reverse@ DeleteCases[Append[Abs@ Differences@ #, Last@ #], k_ /; k == 0] &@ NestWhileList[# - SelectFirst[# - Range[0, # - 1], Block[{m = #, n = 6}, While[And[m != 1, ! CoprimeQ[m, n]], n = GCD[m, n]; m = m/n]; m == 1] &] &, n, # > 1 &], {n, 49}]

from itertools import count, takewhile
N = 50
def B(p): return list(takewhile(lambda x: x<=N, (p**i for i in count(0))))
B23set = set(b*t for b in B(2) for t in B(3) if b*t <= N)
B23lst = sorted(B23set, reverse=True)
def row(n):
    if n in B23set: return [n]
    big = next(t for t in B23lst if t <= n)
    return row(n - big) + [big]
print([t for r in range(1, N) for t in row(r)]) # Michael S. Branicky, Sep 14 2022

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

Original entry on oeis.org

1, 3, 4, 5, 1, 12, 1, 9, 10, 3, 1, 20, 1, 3, 4, 17, 1, 30, 1, 5, 4, 3, 1, 36, 1, 3, 28, 5, 1, 12, 1, 33, 4, 3, 1, 50, 1, 3, 4, 9, 1, 12, 1, 5, 10, 3, 1, 68, 1, 3, 4, 5, 1, 84, 1, 9, 4, 3, 1, 20, 1, 3, 10, 65, 1, 12, 1, 5, 4, 3, 1, 90, 1, 3, 4, 5, 1, 12, 1, 17

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A382488(n), and the largest of them is A065331(n).

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

The unitary analog of A072079.
Cf. A001620, A003586, A034448, A065331, A082633, A382488.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), this sequence (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[n_, p_] := If[Divisible[n, p], p^IntegerExponent[n, p] + 1, 1]; a[n_] := f[n, 2]*f[n, 3]; Array[a, 100]

a(n) = if(n%2, 1, 2^valuation(n, 2)+1) * if(!(n%3), 3^valuation(n, 3)+1, 1);

Multiplicative with a(p^e) = p^e + 1 if p <= 3, and 1 if p >= 5.
a(n) = A034448(n)/A385045(n).
a(n) <= A034448(n), with equality if and only if n 3-smooth.
a(n) <= A072079(n).
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))) * ((1-1/3^(2*s-1))/(1-1/3^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(6*log(2)*log(3))) * (log(n)^2 + c1*log(n) + c2), where c1 = 2*gamma - 2 + 7*log(2) + 5*log(3) - 2*log(6) = 5.916004..., c2 = 2 - 5*log(2) - 11*log(2)^2/6 - 3*log(3) - 5*log(3)^2/6 + 15*log(2)*log(3)/2 + (5*log(2) + 3*log(3) - 2)*gamma - 2*gamma_1 = 1.957142..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633).

A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A003586, A064987, A072044, A072045, A072079.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385138(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (3/4)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.

A202821 Position of 6^n among 3-smooth numbers A003586.

Original entry on oeis.org

1, 5, 14, 26, 43, 64, 89, 119, 153, 191, 233, 279, 330, 385, 444, 507, 575, 646, 722, 802, 886, 975, 1067, 1164, 1266, 1371, 1481, 1595, 1713, 1835, 1961, 2092, 2227, 2366, 2509, 2657, 2809, 2965, 3125, 3289, 3458, 3630, 3807, 3989, 4174, 4364, 4558, 4756

Offset: 0

Zak Seidov, Dec 25 2011

nonn

			a(0) = 1 because A003586(1) = 6^0 = 1.
a(1) = 5 because A003586(5) = 6^1 = 6.
a(2) = 14 because A003586(14) = 6^2 = 36.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A000400, A003586, A022330, A022331.

a[n_] := Sum[Floor[Log[3, 6^n/2^i]] + 1, {i, 0, Log2[6^n]}]; Array[a, 50, 0] (* Amiram Eldar, Jul 15 2023 *)

# uses imports/function in A372401
print(list(islice(A372401gen(p=3), 1000))) # Michael S. Branicky, Jun 06 2024

from sympy import integer_log
def A202821(n): return 1+n*(n+1)+sum((m:=3**i).bit_length()+((1<Chai Wah Wu, Oct 22 2024

A003586(a(n)) = 6^n, for n >= 0.

a(n) ~ (log(6))^2/(log(3)*log(4))*n^2 = 2.1079...*n^2.

cp's OEIS Frontend

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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A022330 Index of 3^n within sequence of numbers of form 2^i*3^j (A003586).

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A022331 Index of 2^n within sequence of numbers of form 2^i*3^j (A003586).

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A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

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A205506 Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

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A276380 Irregular triangle where row n contains terms k of the partition of n produced by greedy algorithm such that all elements are in A003586.

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