A069352 -id:A069352 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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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