A025634 -id:A025634 - cp's OEIS frontend

A025635 Numbers of form 9^i*10^j, with i, j >= 0.

1, 9, 10, 81, 90, 100, 729, 810, 900, 1000, 6561, 7290, 8100, 9000, 10000, 59049, 65610, 72900, 81000, 90000, 100000, 531441, 590490, 656100, 729000, 810000, 900000, 1000000, 4782969, 5314410, 5904900, 6561000, 7290000, 8100000, 9000000

Offset: 1

David W. Wilson

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

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.

import Data.Set (singleton, deleteFindMin, insert)
a025635 n = a025635_list !! (n-1)
a025635_list = f $ singleton (1,0,0) where
   f s = y : f (insert (9 * y, i + 1, j) $ insert (10 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

With[{max = 10^7}, Flatten[Table[9^i*10^j, {i, 0, Log[9, max]}, {j, 0, Log[10, max/9^i]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=9)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

from sympy import integer_log
def A025635(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//10**i,9)[0]+1 for i in range(integer_log(x,10)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Sum_{n>=1} 1/a(n) = 5/4. - Amiram Eldar, Mar 29 2025

a(n) = 9^A025683(n) * 10^A025691(n). - R. J. Mathar, Jul 06 2025

A108779 Numbers of the form (10^i)*(11^j), with i, j >= 0.

Original entry on oeis.org

1, 10, 11, 100, 110, 121, 1000, 1100, 1210, 1331, 10000, 11000, 12100, 13310, 14641, 100000, 110000, 121000, 133100, 146410, 161051, 1000000, 1100000, 1210000, 1331000, 1464100, 1610510, 1771561, 10000000, 11000000, 12100000, 13310000

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 26 2005

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

Cf. A025612, A025616, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108090.

import Data.Set (singleton, deleteFindMin, insert)
a108779 n = a108779_list !! (n-1)
a108779_list = f $ singleton (1,0,0) where
   f s = y : f (insert (10 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

n = 10^7; Flatten[Table[10^i*11^j, {i, 0, Log10[n]}, {j, 0, Log[11, n/10^i]}]] // Sort (* Amiram Eldar, Sep 25 2020 *)

Sum_{n>=1} 1/a(n) = (10*11)/((10-1)*(11-1)) = 11/9. - Amiram Eldar, Sep 25 2020

a(n) ~ exp(sqrt(2*log(10)*log(11)*n)) / sqrt(110). - Vaclav Kotesovec, Sep 25 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

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

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

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

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 *)

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

A025677 Exponent of 8 (value of i) in n-th number of form 8^i*10^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, 11, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 12, 1, 11, 0, 10, 9, 8, 7, 6, 5, 4, 3, 13, 2, 12, 1, 11, 0, 10, 9, 8, 7, 6, 5, 4, 14

Offset: 1

David W. Wilson

nonn

Cf. A025634.

Take[Sort[Flatten[Table[{8^i 10^j,i},{i,0,20},{j,0,20}],1]][[;;,2]],110] (* Harvey P. Dale, Jul 22 2025 *)

A025690 Exponent of 10 (value of j) in n-th number of form 8^i*10^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, 0, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 1, 11, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 1, 11, 2, 12, 3, 4, 5, 6, 7, 8, 9, 0, 10, 1

Offset: 1

David W. Wilson

nonn

Cf. A025634.

cp's OEIS Frontend

A025635 Numbers of form 9^i*10^j, with i, j >= 0.

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A108779 Numbers of the form (10^i)*(11^j), with i, j >= 0.

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

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A025677 Exponent of 8 (value of i) in n-th number of form 8^i*10^j.

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Mathematica

A025690 Exponent of 10 (value of j) in n-th number of form 8^i*10^j.

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