A003586 -id:A003586 - cp's OEIS frontend

A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

1, 2, 6, 12, 24, 72, 144, 288, 864, 1728, 5184, 10368, 20736, 62208, 124416, 373248, 746496, 1492992, 4478976, 8957952, 26873856, 53747712, 107495424, 322486272, 644972544, 1289945088, 3869835264, 7739670528, 23219011584, 46438023168, 92876046336, 278628139008, 557256278016

Offset: 1

Amiram Eldar, Jul 06 2019

nonn

How is this related to A163895? - R. J. Mathar, May 05 2023

			From _Michael De Vlieger_, Jul 12 2019: (Start)
We can plot all terms in A003586 with the power range 2^x with x >= 0 and 3^y with y >= 0 on the x and y axis, respectively. Plot of terms m in A309015, with terms also in a(n) placed in brackets:
                                2^x
          0    1     2     3     4     5     6     7     8
        +-----------------------------------------------------
     0  |[1]  [2]    4
     1  |     [6]  [12]  [24]   48
3^y  2  |           36   [72] [144]  [288]   576
     3  |                216   432   [864] [1728] 3456  6912 ...
          ...
Larger scale plot with "." representing a term m in A309015, and "o" representing a term in A309015 also in a(n) for all m < A002110(20).
                              2^x
        0    5   10   15   20   25   30   35   40   45  ...
        +------------------------------------------------
       0|oo.
        | ooo.
        |  .ooo.
        |   ..oo..
        |    ..ooo..
       5|      ..oo...
        |       ..ooo...
        |         ..oo....
        |          ..ooo....
        |            ..ooo....
      10|             ...oo.....
        |               ..ooo....
        |                ...oo.....
        |                  ..ooo.....
3^y     |                   ...ooo....
      15|                     ...oo.....
        |                      ...ooo.....
        |                        ...oo.....
        |                         ...ooo.....
        |                           ...oo......
      20|                            ...ooo.....
        |                              ...ooo.....
        |                               ....oo......
        |                                 ...ooo.....
        |                                  ....oo......
      25|                                    ...ooo......
        |                                     ....ooo....
        |                                       ....oo.
        |                                        ....o
        |                                          .
     ...
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..2709
Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
Michael De Vlieger, Factors p analogous to A000705 such that the product of the smallest n terms equals a(n + 1) (10^5 terms).

Subsequence of A003586 and A309015.

Cf. A000005, A002201.

f[nn_, k_: 2] := Block[{w = {{2, 1}, {3, 0}}, s = {2}, P = 1, q = k - 2, x, i, n, f}, f[w_List] := Log[#1, (#2 + 2)/(#2 + 1)] & @@ w; x = Array[f[w[[#]] ] &, P + 1]; For[n = 2, n <= nn, n++, i = First@ FirstPosition[x, Max[x]]; AppendTo[s, w[[i, 1]]]; w[[i, 2]]++; If[And[i > P, P <= q], P++; AppendTo[w, {Prime[i + 1], 0}]; AppendTo[x, f[Last@ w]]]; x[[i]] = f@ w[[i]] ]; s]; {1}~Join~FoldList[Times, f[32, 2]] (* Michael De Vlieger, Jul 11 2019, after T. D. Noe at A000705 *)

More terms from Michael De Vlieger, Jul 11 2019

A352072 a(n) = least k such that A003586(n) | 12^k.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Mar 08 2022

Also, number of digits in the duodecimal expansion of terminating unit fractions 1/A003586.

			a(1) = 0 since A003586(1) = 1 | 12^0.
a(2) = 1 since A003586(2) = 2 | 12^1; 1/2 expanded in base 12 = .6.
a(6) = 2 since A003586(6) = 8 | 12^2; 1/8 in base 12 = .16.
a(12) = 3 since A003586(12) = 27 | 12^3; 1/27 in base 12 = .054, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 1..10283 (A003586(10283) = 12^50)
Eric Weisstein's World of Mathematics, Duodecimal.
Wikipedia, Duodecimal.

Cf. A003586, A086415, A117920.

With[{nn = 40000}, Sort[Join @@ Table[{2^a*3^b, Max[Ceiling[a/2], b]}, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}] ][[All, -1]] ]

A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 2, 4, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The sum of these divisors is A365209(n).

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

Cf. A000005, A003586, A034444, A065330, A065331, A365209.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3, e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, f[i,2]+1, 2));}

Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000005(A065331(n)) * A034444(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).

A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 26, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 50, 78, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The number of these divisors is A365208(n).

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

Cf. A000203, A003586, A034448, A065330, A065331, A365208.

Cf. A002117, A013661, A306633.

f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1 + f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000203(A065331(n)) * A034448(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .

A098229 a(n) = 6c(m,1) where m = A003586(n) is the n-th 3-smooth number, c(m,k) = {(m^(2k)-1)B(2k)}, {x} denotes the fractional part of x and B(k) is the k-th Bernoulli number.

Original entry on oeis.org

0, 3, 2, 3, 5, 3, 2, 5, 3, 5, 5, 2, 3, 5, 5, 5, 3, 5, 2, 5, 5, 3, 5, 5, 5, 5, 2, 3, 5, 5, 5, 5, 5, 3, 5, 5, 2, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 3, 2, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 2, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 2, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Benoit Cloitre, Oct 25 2004

nonn

If m is a 3-smooth number (i.e., of form 2^i*3^j for i,j >= 0), the value of c(m,k) is independent of k.

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

Cf. A003586, A027641, A027642.

s[n_] := If[Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]) == n, 6 * FractionalPart[(n^2-1)/6], Nothing]; Array[s, 125000] (* Amiram Eldar, May 03 2025 *)

m=7;for(n=1,1000000,if(gcd(n,6^100)==n,print1(6*frac((n^(2*m)-1)*bernfrac(2*m)),",")))

a(1) = 0; for k > 0, a(2^k) = 3 and a(3^k) = 2; for i > 0 and j > 0, a(2^i*3^j) = 5.

A317687 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms is 3-smooth (A003586).

Original entry on oeis.org

1, 2, 4, 5, 3, 6, 10, 8, 16, 11, 7, 9, 15, 12, 20, 28, 26, 22, 14, 13, 19, 17, 31, 23, 25, 29, 35, 37, 27, 21, 33, 39, 42, 30, 18, 36, 45, 51, 57, 24, 40, 32, 49, 47, 34, 38, 43, 53, 55, 41, 67, 61, 83, 79, 65, 63, 81, 111, 105, 87, 75, 69, 59, 85, 77, 115

Offset: 1

Rémy Sigrist, Aug 04 2018

nonn

The 2-smooth variant of this sequence, say b, satisfies b(n) = A001045(n+1) for any n > 0.

			The first terms, alongside a(n) + a(n+1), are:
  n   a(n)    a(n) + a(n+1)
  --  ----    -------------
   1     1    3
   2     2    2 * 3
   3     4    3^2
   4     5    2^3
   5     3    3^2
   6     6    2^4
   7    10    2 * 3^2
   8     8    2^3 * 3
   9    16    3^3
  10    11    2 * 3^2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 6500000 terms
Rémy Sigrist, C++ program for A317687

Cf. A001045, A003586.

Nest[Append[#, Block[{k = 2}, While[Nand[FreeQ[#, k], SubsetQ[{2, 3}, FactorInteger[k + #[[-1]] ][[All, 1]]]], k++]; k]] &, {1}, 65] (* Michael De Vlieger, Aug 05 2018 *)

A353383 Irregular triangle T(n,k) with row n listing A003586(j) not divisible by 12 such that A352072(A003586(j)) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 16, 18, 27, 32, 54, 64, 81, 128, 162, 256, 243, 486, 512, 1024, 729, 1458, 2048, 4096, 2187, 4374, 8192, 16384, 6561, 13122, 32768, 65536, 19683, 39366, 131072, 262144, 59049, 118098, 524288, 1048576, 177147, 354294, 2097152, 4194304, 531441, 1062882, 8388608, 16777216

Offset: 0

Michael De Vlieger, Apr 15 2022

All terms in A003586 are products T(n,k)*12^j, j >= 0.
When expressed in base 12, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 12.
The first 5 terms are the proper divisors of 12.
For these reasons, the terms may be called duodecimal "proper regular" numbers.

			Row 0 contains 1 since 1 is the empty product.
Row 1 contains 2, 3, 4, and 6 since these divide 12.
Row 2 contains 8, 9, 16, and 18 since these divide 12^2 but not 12. The other divisors of 12^2 either divide smaller powers of 12 or they are divisible by 12 and do not appear.
Row 3 contains 27, 32, 54, and 64 since these divide 12^3 but not 12^2. The other divisors of 12^3 either divide smaller powers of 12 or they are divisible by 12 therefore do not appear.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.

Michael De Vlieger, Table of n, a(n) for n = 0..6643
Eric Weisstein's World of Mathematics, Duodecimal.
Wikipedia, Duodecimal.

Cf. A003586, A352072.

{{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 # - 1), 3^#} &, 12] // Flatten

Row 0 contains the empty product, thus row length = 1.

Row n sorts {2^(2n-1), 3^n, 2^(2n), 2*3^n}, thus row length = 4.

A372744 If the n-th 3-smooth number, A003586(n), equals 2^i * 3^j for some i, j >= 0, then the a(n)-th 3-smooth number, A003586(a(n)), equals 2^j * 3^i.

Original entry on oeis.org

1, 3, 2, 7, 5, 12, 4, 10, 19, 8, 16, 6, 27, 14, 24, 11, 37, 21, 9, 33, 18, 49, 30, 15, 44, 26, 13, 62, 40, 23, 57, 36, 20, 77, 52, 32, 17, 71, 47, 29, 93, 66, 43, 25, 87, 60, 39, 111, 22, 81, 55, 35, 104, 75, 51, 131, 31, 98, 69, 46, 123, 28, 91, 64, 152, 42

Offset: 1

Rémy Sigrist, May 12 2024

nonn

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points (A202821).

			A003586(8) = 12 = 2^2 * 3^1, A003586(10) = 18 = 2^1 * 3^2, so a(8) = 10 and.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A003586, A022328, A022329, A202821 (fixed points).

PARI
```
\\ See Links section.
```

A022328(a(n)) = A022329(n).
A022329(a(n)) = A022328(n).
a(n) = n iff n belongs to A202821.
sign(a(n) - n) = sign(A022328(n) - A022329(n)).

A374484 Index of A006899(n) in A003586.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 12, 13, 17, 19, 22, 27, 28, 34, 37, 41, 48, 49, 56, 62, 65, 74, 77, 84, 93, 95, 106, 111, 118, 130, 131, 143, 152, 157, 171, 175, 186, 199, 202, 218, 225, 235, 252, 253, 271, 281, 290, 309, 312, 329, 344, 350, 371, 378, 393, 413, 416, 439

Offset: 1

Chai Wah Wu, Sep 16 2024

Index of powers of 2 and 3 in 3-smooth numbers.

			A006899(10) = 64 which is the 17th term of A003586, therefore a(10) = 17.

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

Cf. A003586, A006899.

Disjoint union of A022330 and A022331.

seq[lim_] := Position[Times @@ IntegerExponent[#, {2, 3}] & /@ Sort[Flatten[ Table[2^i*3^j, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}] ]], 0] // Flatten; seq[10^11] (* Amiram Eldar, Sep 18 2024 *)

from sympy import integer_log
def A374484(n): return sum(((1<

A003586(a(n)) = A006899(n).

a(n) ~ c * n^2, where c = log(2)*log(3)/(2*(log(2) + log(3))^2) = 0.118598856384648... - Vaclav Kotesovec and Amiram Eldar, Sep 19 2024

A277045 Irregular triangle T(n,k) read by rows giving the number of partitions of length k such that all of the members of the partition are distinct and in A003586.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Sep 27 2016

If n is in A003586, then T(n,1) = 1, else T(n,1) = 0.
T(n,k) also is the number of ways of representing n involving k 1's in the base(2,3) or "dual-base number system" (i.e., base(2,3)).
The number of "canonic" representations of n in a dual-base number system as defined by the reference as having the lowest number of terms, appears in the first column of the triangle with a value greater than 0.
A237442(n) = the least k with a nonzero value.

			Triangle starts:
1
1
1,1
1,1
0,2
1,1,1
0,2,1
1,1,1
1,2,2
0,3,1,1
0,2,3
1,2,3,1
0,2,4,1
0,2,3,2
0,2,4,3
1,1,4,2,1
0,2,4,3
1,2,4,4,1
0,2,5,4,1
0,3,3,5,1
...
Row n = 10 has terms {0,3,1,1} because 10 is not in A003586 thus k = 1 has value 0. The partitions of 10 that have distinct members that are in A003586 are {{1,9},{2,8},{4,6},{1,3,6},{1,2,3,4}}, thus there are 3 partitions of length k = 2, 1 of length k = 3, and 1 with k = 4. A237442(10) = 2.

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

Cf. A003586, A237442, A276380.

nn = 6^6; t = Sort@ Select[Flatten@ KroneckerProduct[2^Range[0, Ceiling@ Log2@ nn], 3^Range[0, Ceiling@ Log[3, nn]]], # <= nn &]; Table[BinCounts[#, {1, Max@ # + 1, 1}] &@ Map[Length, #] &@ Select[Subsets@ TakeWhile[t, # <= n &], Total@ # == n &], {n, 40}]

cp's OEIS Frontend

A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

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A352072 a(n) = least k such that A003586(n) | 12^k.

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A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

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A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

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A098229 a(n) = 6*c(m,1) where m = A003586(n) is the n-th 3-smooth number, c(m,k) = {(m^(2*k)-1)*B(2*k)}, {x} denotes the fractional part of x and B(k) is the k-th Bernoulli number.

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A317687 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms is 3-smooth (A003586).

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Mathematica

A353383 Irregular triangle T(n,k) with row n listing A003586(j) not divisible by 12 such that A352072(A003586(j)) = n.

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A372744 If the n-th 3-smooth number, A003586(n), equals 2^i * 3^j for some i, j >= 0, then the a(n)-th 3-smooth number, A003586(a(n)), equals 2^j * 3^i.

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A098229 a(n) = 6c(m,1) where m = A003586(n) is the n-th 3-smooth number, c(m,k) = {(m^(2k)-1)B(2k)}, {x} denotes the fractional part of x and B(k) is the k-th Bernoulli number.