A022329 -id:A022329 - cp's OEIS frontend

A086417 Sum of divisors of 3-smooth numbers.

1, 3, 4, 7, 12, 15, 13, 28, 31, 39, 60, 40, 63, 91, 124, 120, 127, 195, 121, 252, 280, 255, 403, 363, 508, 600, 364, 511, 819, 847, 1020, 1240, 1092, 1023, 1651, 1815, 1093, 2044, 2520, 2548, 2047, 3315, 3751, 3279, 4092, 5080, 5460, 4095, 3280

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A086416.

Cf. A000203, A003586, A022328, A022329.

a086417 n = (2 ^ (a022328 n + 1) - 1) * (3 ^ (a022329 n + 1) - 1) `div` 2
-- Reinhard Zumkeller, Nov 19 2015

DivisorSigma[1,#]&/@Select[Range[4000],Max[FactorInteger[#][[All,1]]]<4&] (* Harvey P. Dale, Feb 14 2017 *)

a(n) = A000203(A003586(n));
a(n) = (2^(A022328(n)+1)-1)*(3^(A022329(n)+1)-1)/2.
a(n) = A000225(j) * A003462(k) for some j,k > 0. - Flávio V. Fernandes, May 29 2021

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

A069357 Numbers of form 2^i*3^j + (i+j) with i, j >= 0.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 15, 20, 21, 28, 30, 37, 40, 53, 58, 70, 77, 85, 102, 113, 135, 150, 167, 199, 222, 248, 264, 295, 330, 392, 439, 492, 521, 584, 655, 735, 777, 872, 979, 1034, 1161, 1304, 1465, 1546, 1737, 1952, 2059, 2194, 2314, 2601, 2924, 3083, 3466, 3897

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Cf. A055600, A069358, A064800, A022328, A022329.

Distinct values of A003586(k) + A069352(k). [Corrected by Georg Fischer, Dec 11 2022, further clarification by Sean A. Irvine, Apr 28 2024]

Missing a(1)=1 inserted and duplicate values removed by Sean A. Irvine, Apr 28 2024

A086419 Sum of all prime factors of 3-smooth numbers.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 21, 22, 22, 22, 23, 23, 23, 24, 23, 24, 24, 24, 25, 24, 25, 25, 26, 25, 26, 26, 26, 27

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A001414, A003586, A022328, A022329, A086418, A086417, A069352.

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

a(n) = A001414(A003586(n)).

a(n) = 2*A022328(n) + 3*A022329(n).

A257999 Numbers of the form, 2^i*3^j, i+j odd.

Original entry on oeis.org

2, 3, 8, 12, 18, 27, 32, 48, 72, 108, 128, 162, 192, 243, 288, 432, 512, 648, 768, 972, 1152, 1458, 1728, 2048, 2187, 2592, 3072, 3888, 4608, 5832, 6912, 8192, 8748, 10368, 12288, 13122, 15552, 18432, 19683, 23328, 27648, 32768, 34992, 41472, 49152, 52488

Offset: 1

Reinhard Zumkeller, May 16 2015

nonn

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

Complement of A036667 with respect to A003586.
Intersection of A026424 and A003586.
Cf. A069352, A022328, A022329.

a257999 n = a257999_list !! (n-1)
a257999_list = filter (odd . flip mod 2 . a001222) a003586_list

max = 53000; Reap[Do[k = 2^i*3^j; If[k <= max && OddQ[i + j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 after Jean-François Alcover at A036667 *)

from sympy import integer_log
def A257999(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 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 = 1.

Sum_{n>=1} 1/a(n) = 5/4. - Amiram Eldar, Feb 18 2021

A067371 Arithmetic derivatives of 3-smooth numbers.

Original entry on oeis.org

0, 1, 1, 4, 5, 12, 6, 16, 32, 21, 44, 27, 80, 60, 112, 81, 192, 156, 108, 272, 216, 448, 384, 297, 640, 540, 405, 1024, 912, 756, 1472, 1296, 1053, 2304, 2112, 1836, 1458, 3328, 3024, 2592, 5120, 4800, 4320, 3645, 7424, 6912, 6156, 11264, 5103, 10752

Offset: 1

Reinhard Zumkeller, Mar 20 2002, revised: Jul 19 2003

nonn

			a(18) = A003415(A003586(18)) = A003415(72) = A003415(2^3*3^2) = (3*3+2*2)*2^(3-1)*3^(2-1) = (9+4)*2^2*3^1 = 13*4*3 = 156.
a(27) = A003415(A003586(27)) = A003415(243) = A003415(2^0*3^5) = (3*0+2*5)*2^(0-1)*3^(5-1) = ((0+10)/2)*3^4 = 5*81 = 405.

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

Cf. A001787, A003415, A003586, A027471, A022328, A022329.

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

A003415(2^i+3^j) = (3*i + 2*j) * 2^(i-1) * 3^(j-1), i, j >=0.

a(n) = A003415(A003586(n)).

A086416 Number of divisors of 3-smooth numbers.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 6, 5, 6, 8, 4, 6, 9, 10, 8, 7, 12, 5, 12, 12, 8, 15, 10, 14, 16, 6, 9, 18, 15, 16, 20, 12, 10, 21, 20, 7, 18, 24, 18, 11, 24, 25, 14, 20, 28, 24, 12, 8, 27, 30, 21, 22, 32, 30, 13, 16, 30, 35, 28, 24, 9, 36, 36, 14, 24, 33, 40, 35, 26, 18, 40, 42, 15, 32

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A086414, A086415, A086417.

Cf. A000005, A003586, A022328, A022329.

DivisorSigma[0, Select[Range[10000], # == 2^IntegerExponent[#, 2] * 3^IntegerExponent[#, 3] &]] (* Amiram Eldar, Apr 15 2024 *)

a(n) = A000005(A003586(n)).
a(n) = if A086414(n) = A086415(n) then A086414(n)+1 else (A086414(n)+1)*(A086415(n)+1).
a(n) = (A022328(n)+1)*(A022329(n)+1).

A086418 Sum of distinct prime factors of 3-smooth numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

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

Cf. A003586, A008472, A022328, A022329, A086419, A086417, A086412.

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

a(n) = A008472(A003586(n));

a(n) = 2*0^(0^A022328(n)) + 3*0^(0^A022329(n)).

A033031 Squarefree kernels of 3-smooth numbers.

Original entry on oeis.org

1, 2, 3, 2, 6, 2, 3, 6, 2, 6, 6, 3, 2, 6, 6, 6, 2, 6, 3, 6, 6, 2, 6, 6, 6, 6, 3, 2, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 2, 3, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 3, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 3, 6, 2, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Jun 21 2003

nonn

			A003586(17) = 64 = 2^6 -> a(17) = 2,
A003586(18) = 72 = 2^3 * 3^2 -> a(18) = 2*3 = 6,
A003586(19) = 81 = 3^4 -> a(19) = 3.

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

Cf. A003586, A007947, A022328, A022329.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); rad /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A007947(A003586(n)).
a(n) = (2*0^(A022328(n)-1)) * (3*0^(A022329(n)-1)) for n>1. - Reinhard Zumkeller, Jul 18 2003
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6. - Amiram Eldar, Jul 13 2023

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)).

cp's OEIS Frontend

A086417 Sum of divisors of 3-smooth numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A069357 Numbers of form 2^i*3^j + (i+j) with i, j >= 0.

Views

Author

Keywords

Crossrefs

Formula

Extensions

A086419 Sum of all prime factors of 3-smooth numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A257999 Numbers of the form, 2^i*3^j, i+j odd.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A067371 Arithmetic derivatives of 3-smooth numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A086416 Number of divisors of 3-smooth numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A086418 Sum of distinct prime factors of 3-smooth numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A033031 Squarefree kernels of 3-smooth numbers.

Views

Author

Keywords