A025620 -id:A025620 - cp's OEIS frontend

A107988 Numbers of the form (4^i)*(11^j), with i, j >= 0.

1, 4, 11, 16, 44, 64, 121, 176, 256, 484, 704, 1024, 1331, 1936, 2816, 4096, 5324, 7744, 11264, 14641, 16384, 21296, 30976, 45056, 58564, 65536, 85184, 123904, 161051, 180224, 234256, 262144, 340736, 495616, 644204, 720896, 937024, 1048576

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 12 2005

nonn

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

Subsequence of A003596.

Cf. A025617, A025618, A025619, A025620, A025621, A107462, A003597, A003598, A003599, A108090.

import Data.Set (singleton, deleteFindMin, insert)
a107988 n = a107988_list !! (n-1)
a107988_list = f $ singleton (1,0,0) where
   f s = y : f (insert (4 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

n = 10^6; Flatten[Table[4^i*11^j, {i, 0, Log[4, n]}, {j, 0, Log[11, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

Sum_{n>=1} 1/a(n) = (4*11)/((4-1)*(11-1)) = 22/15. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(4)*log(11)*n)) / sqrt(44). - Vaclav Kotesovec, Sep 24 2020

A107462 Numbers of the form (4^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 4, 13, 16, 52, 64, 169, 208, 256, 676, 832, 1024, 2197, 2704, 3328, 4096, 8788, 10816, 13312, 16384, 28561, 35152, 43264, 53248, 65536, 114244, 140608, 173056, 212992, 262144, 371293, 456976, 562432, 692224, 851968, 1048576, 1485172

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 09 2005

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

Subsequence of A107326.

Cf. A025617, A025618, A025619, A025620, A025621, A107364, A107466, A108056, A108090.

n = 10^6; Flatten[Table[4^i*13^j, {i, 0, Log[4, n]}, {j, 0, Log[13, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)

Sum_{n>=1} 1/a(n) = (4*13)/((4-1)*(13-1)) = 13/9. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(4)*log(13)*n)) / sqrt(52). - Vaclav Kotesovec, Sep 24 2020

A036667 Numbers of the form 2^i*3^j, i+j even.

Original entry on oeis.org

1, 4, 6, 9, 16, 24, 36, 54, 64, 81, 96, 144, 216, 256, 324, 384, 486, 576, 729, 864, 1024, 1296, 1536, 1944, 2304, 2916, 3456, 4096, 4374, 5184, 6144, 6561, 7776, 9216, 11664, 13824, 16384, 17496, 20736, 24576, 26244, 31104, 36864, 39366

Offset: 1

N. J. A. Sloane

nonn

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

Complement of A257999 with respect to A003586.
Intersection of A028260 and A003586.
Cf. A025620 (subsequence), A069352, A022328, A022329.

a036667 n = a036667_list !! (n-1)
a036667_list = filter (even . flip mod 2 . a001222) a003586_list
-- Reinhard Zumkeller, May 16 2015

max = 40000;
Reap[Do[k = 2^i 3^j; If[k <= max && EvenQ[i+j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}]][[2, 1]] // Union (* Jean-François Alcover, Aug 04 2018 *)

from sympy import integer_log
def A036667(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)>>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 = 0. - Reinhard Zumkeller, May 16 2015

Sum_{n>=1} 1/a(n) = 7/4. - Amiram Eldar, Feb 18 2021

Offset corrected by Reinhard Zumkeller, May 16 2015

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A107988 Numbers of the form (4^i)*(11^j), with i, j >= 0.

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A107462 Numbers of the form (4^i)*(13^j), with i, j >= 0.

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A036667 Numbers of the form 2^i*3^j, i+j even.

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