cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A107788 Numbers of the form (8^i)*(11^j), with i, j >= 0.

Original entry on oeis.org

1, 8, 11, 64, 88, 121, 512, 704, 968, 1331, 4096, 5632, 7744, 10648, 14641, 32768, 45056, 61952, 85184, 117128, 161051, 262144, 360448, 495616, 681472, 937024, 1288408, 1771561, 2097152, 2883584, 3964928, 5451776, 7496192, 10307264
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 14 2005

Keywords

Links

Crossrefs

Cf. A025633, A025634, A107764, A003596, A003597, A107988, A003598, A003599, A108090.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a107788 n = a107788_list !! (n-1)
a107788_list = f $ singleton (1,0,0) where
   f s = y : f (insert (8 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015
  • Mathematica
    Take[Union[8^First[#]*11^Last[#]&/@Tuples[Range[0,20],2]],40] (* Harvey P. Dale, Jan 17 2015 *)
n = 10^6; Flatten[Table[8^i*11^j, {i, 0, Log[8, n]}, {j, 0, Log[11, n/8^i]}]] // Sort (* Amiram Eldar, Oct 07 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (8*11)/((8-1)*(11-1)) = 44/35. - Amiram Eldar, Oct 07 2020
a(n) ~ exp(sqrt(2*log(8)*log(11)*n)) / sqrt(88). - Vaclav Kotesovec, Oct 07 2020

A108687 Numbers of the form (9^i)*(11^j), with i, j >= 0.

Original entry on oeis.org

1, 9, 11, 81, 99, 121, 729, 891, 1089, 1331, 6561, 8019, 9801, 11979, 14641, 59049, 72171, 88209, 107811, 131769, 161051, 531441, 649539, 793881, 970299, 1185921, 1449459, 1771561, 4782969, 5845851, 7144929, 8732691, 10673289, 13045131
Offset: 1

Views

Author

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

Keywords

Links

Crossrefs

Subsequence of A003597.
Cf. A025633, A025634, A107788, A107764, A003596, A107988, A003598, A003599, A108090.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a108687 n = a108687_list !! (n-1)
a108687_list = f $ singleton (1,0,0) where
   f s = y : f (insert (9 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015
  • Mathematica
    f[upto_]:=With[{max9=Floor[Log[9,upto]],max11=Floor[Log[11,upto]]}, Select[Union[Times@@{9^First[#],11^Last[#]}&/@Tuples[{Range[0, max9], Range[0, max11]}]], #<=upto&]]; f[14000000]  (* Harvey P. Dale, Mar 11 2011 *)
  • Python
    from sympy import integer_log
def A108687(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//11**i,9)[0]+1 for i in range(integer_log(x,11)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Formula

Sum_{n>=1} 1/a(n) = (9*11)/((9-1)*(11-1)) = 99/80. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(2*log(9)*log(11)*n)) / sqrt(99). - Vaclav Kotesovec, Sep 24 2020

A107764 Numbers of the form (8^i)*(13^j), with i, j >= 0.

Original entry on oeis.org

1, 8, 13, 64, 104, 169, 512, 832, 1352, 2197, 4096, 6656, 10816, 17576, 28561, 32768, 53248, 86528, 140608, 228488, 262144, 371293, 425984, 692224, 1124864, 1827904, 2097152, 2970344, 3407872, 4826809, 5537792, 8998912, 14623232, 16777216
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 11 2005

Keywords

Links

Crossrefs

Cf. A025633, A025634, A107326, A107364, A107462, A107466, A107710, A108056, A108090.

Programs

  • Mathematica
    n = 10^6; Flatten[Table[8^i*13^j, {i, 0, Log[8, n]}, {j, 0, Log[13, n/8^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Formula

Sum_{n>=1} 1/a(n) = (8*13)/((8-1)*(13-1)) = 26/21. - Amiram Eldar, Sep 25 2020
a(n) ~ exp(sqrt(2*log(8)*log(13)*n)) / sqrt(104). - Vaclav Kotesovec, Sep 25 2020

A025682 Exponent of 9 (value of j) in n-th number of form 8^i*9^j.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Cf. A025633. Differs from A002262 at a(190).

A025676 Exponent of 8 (value of i) in n-th number of form 8^i*9^j.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3
Offset: 1

Views

Author

David W. Wilson

Keywords

Crossrefs

Cf. A025633, A025682. Differs from A025581 at a(190).

A025732 Index of 8^n within sequence of numbers of form 8^i*9^j.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 190, 209, 229, 250, 272, 295, 319, 344, 370, 397, 425, 454, 484, 515, 547, 580, 614, 649, 685, 721, 758, 796, 835, 875, 916, 958, 1001, 1045, 1090, 1136, 1183, 1231, 1280, 1330, 1381, 1433
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Positions of zeros in A025682. - R. J. Mathar, Jul 06 2025

Crossrefs

Cf. A001018, A025633.
Showing 1-6 of 6 results.

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