A022328 -id:A022328 - cp's OEIS frontend

A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 15, 19, 20, 21, 27, 32, 36, 37, 47, 48, 54, 64, 65, 80, 85, 92, 112, 113, 114, 135, 150, 158, 193, 199, 200, 228, 263, 264, 273, 329, 350, 351, 387, 457, 464, 474, 558, 614, 615, 616, 661, 787, 815, 826, 946, 1072, 1073, 1081, 1136

Offset: 0

Henry Bottomley, May 24 2001

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A061984, A061986, A061987.

a061985 n = a061985_list !! n
a061985_list = f (-1) a061984_list where
   f u (v:vs) = if v == u then f u vs else v : f v vs
-- Reinhard Zumkeller, Jan 11 2014

a(n) = a(n-1) + C(A022328(n) + A022329(n), A022328(n)). - David Wasserman, Nov 17 2005

A086417 Sum of divisors of 3-smooth numbers.

Original entry on oeis.org

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)).

A258051 Fractal sequence derived from A258033.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 2, 4, 1, 3, 0, 5, 2, 4, 1, 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, 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, 3, 11, 0, 8, 5, 13, 2, 10, 7, 15, 4

Offset: 1

Clark Kimberling and Reinhard Zumkeller, May 17 2015

nonn

The sequence is constructed like A258033 is constructed: after partitioning A258033 into segments starting with 0, in each segment the greatest term is to be deleted (see example);

this sequence is fractal, i.e. if the first occurrence of each n is removed, the resulting sequence is the original sequence.

			Segments of A258033 starting with 0, deleted maxima in brackets:
.   1:  0
.   2:  0 [2] 1
.   3:  0 2 1 [3]
.   4:  0 [5] 2 4 1 3
.   5:  0 5 2 4 1 [6] 3
.   6:  0 [8] 5 2 7 4 1 6 3
.   7:  0 8 5 2 [10] 7 4 1 9 6 3
.   8:  0 8 5 2 10 7 4 1 9 6 3 [11]
.   9:  0 8 5 [13] 2 10 7 4 12 1 9 6 3 11
.  10:  0 8 5 13 2 10 7 4 12 1 9 6 [14] 3 11
.  11:  0 8 [16] 5 13 2 10 7 15 4 12 1 9 6 14 3 11
.  12:  0 8 16 5 13 2 10 [18] 7 15 4 12 1 9 17 6 14 3 11
.  13:  0 8 16 5 13 2 10 18 7 15 4 12 1 9 17 6 14 3 11 [19]
.  14:  0 8 16 5 13 [21] 2 10 18 7 15 4 12 20 1 9 17 6 14 3 11 19
.  15:  0 8 16 5 13 21 2 10 18 7 15 4 12 20 1 9 17 6 14 [22] 3 11 19

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

%Cf. A258033, A022328.

import Data.List (delete)
a258051 n = a258051_list !! (n-1)
a258051_list = f (tail a258033_list) where
   f xs = (0 : (delete (maximum ys) ys)) ++ f zs
          where (ys, (_ : zs)) = span (> 0) xs

cp's OEIS Frontend

A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

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A086417 Sum of divisors of 3-smooth numbers.

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

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Mathematica

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

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

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

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A067371 Arithmetic derivatives of 3-smooth numbers.

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A086416 Number of divisors of 3-smooth numbers.

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

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