A108201 -id:A108201 - cp's OEIS frontend

A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

1, 7, 12, 49, 84, 144, 343, 588, 1008, 1728, 2401, 4116, 7056, 12096, 16807, 20736, 28812, 49392, 84672, 117649, 145152, 201684, 248832, 345744, 592704, 823543, 1016064, 1411788, 1741824, 2420208, 2985984, 4148928, 5764801, 7112448

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 17 2005

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

Cf. A036566, A025631, A025632, A003599, A108056, A108201, A064476.

Take[7^#[[1]]*12^#[[2]]&/@Tuples[Range[0,10],2]//Union,40] (* Harvey P. Dale, Mar 05 2017 *)
n = 10^6; Flatten[Table[7^i*12^j, {i, 0, Log[7, n]}, {j, 0, Log[12, n/7^i]}]] // Sort (* Amiram Eldar, Sep 26 2020 *)

from sympy import integer_log
def A108238(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(integer_log(x//12**i,7)[0]+1 for i in range(integer_log(x,12)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 26 2025

Sum_{n>=1} 1/a(n) = (7*12)/((7-1)*(12-1)) = 14/11. - Amiram Eldar, Sep 26 2020

a(n) ~ exp(sqrt(2*log(7)*log(12)*n)) / sqrt(84). - Vaclav Kotesovec, Sep 26 2020

A317804 Numbers of form 2^i*12^j, with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 36864, 41472, 49152, 55296, 65536

Offset: 1

Dario Ch, Sep 01 2018

Dario Ch, Table of n, a(n) for n = 1..10000

Cf. A025612, A003596, A107326, A003597, A107364, A025616, A108201, A108238.

With[{max = 10^5}, Flatten[Table[2^i*12^j, {i, 0, Log2[max]}, {j, 0, Log[12, max/2^i]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

from heapq import heappush, heappop
def sequence():
    pq = [1]
    seen = set(pq)
    while True:
        value = heappop(pq)
        yield value
        seen.remove(value)
        for x in 2 * value, 12 * value:
            if x not in seen:
                heappush(pq, x)
                seen.add(x)
seq = sequence()
finalsequence_list = [next(seq) for i in range(100)]  # Dario Ch, Sep 01 2018

from sympy import integer_log
def A317804(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//12**i).bit_length() for i in range(integer_log(x,12)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 26 2025

Sum_{n>=1} 1/a(n) = 24/11. - Amiram Eldar, Mar 29 2025

A108218 Numbers of the form (11^i)*(12^j), with i, j >= 0.

Original entry on oeis.org

1, 11, 12, 121, 132, 144, 1331, 1452, 1584, 1728, 14641, 15972, 17424, 19008, 20736, 161051, 175692, 191664, 209088, 228096, 248832, 1771561, 1932612, 2108304, 2299968, 2509056, 2737152, 2985984, 19487171, 21258732, 23191344, 25299648

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 28 2005

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

Cf. A003596, A003597, A107988, A003598, A108201, A108698, A064476, A003599, A108238, A107788, A108687, A108779, A108090, A108771.

import Data.Set (singleton, deleteFindMin, insert)
a108218 n = a108218_list !! (n-1)
a108218_list = f $ singleton (1,0,0) where
   f s = y : f (insert (11 * y, i + 1, j) $ insert (12 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

With[{max = 3*10^7}, Flatten[Table[11^i*12^j, {i, 0, Log[11, max]}, {j, 0, Log[12, max/11^i]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

Sum_{n>=1} 1/a(n) = 6/5. - Amiram Eldar, Mar 29 2025

cp's OEIS Frontend

A108238 Numbers of the form (7^i)*(12^j), with i, j >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A317804 Numbers of form 2^i*12^j, with i, j >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

A108218 Numbers of the form (11^i)*(12^j), with i, j >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula