A025612 -id:A025612 - cp's OEIS frontend

A032533 Numbers that, when expressed in base 2 and then interpreted in base 10, yield a multiple of the original number.

1, 2, 4, 8, 10, 16, 20, 21, 32, 40, 42, 64, 80, 84, 100, 128, 160, 168, 200, 256, 273, 320, 336, 400, 512, 546, 640, 672, 800, 1000, 1024, 1092, 1280, 1344, 1600, 2000, 2048, 2184, 2231, 2510, 2560, 2688, 2730, 3200, 3300, 4000, 4096, 4368, 4462

Offset: 1

Patrick De Geest, Apr 15 1998

nonn

Note that A025612 is a subset of this sequence (numbers of form 2^i*10^j, with i, j >= 0).

			8 in base 2 is 1000, which interpreted in base 10 is 1000 = 125*8.

Giovanni Resta, Table of n, a(n) for n = 1..1212 (terms < 3*10^11, first 251 terms from Paolo P. Lava)

Cf. A032532, A032534, A007088.

[k:k in [1..5000]| Seqint(Intseq(k,2)) mod k eq 0]; // Marius A. Burtea, Oct 11 2019

Select[Range[10000], Mod[FromDigits[IntegerDigits[#, 2]], #] == 0 &] (* Carl Najafi, Aug 18 2011 *)

select( is_A032533(n)=fromdigits(binary(n))%n==0, [1..5000]) \\ M. F. Hasler, Oct 11 2019

A032533 = { m : m divides A007088(m) }. - M. F. Hasler, Oct 11 2019

Example and better description from Erich Friedman, Jul 21 2001
Edited by Erich Friedman, Feb 09 2002
Offset set to 1 by Giovanni Resta, Jul 13 2016
Name edited by Jon E. Schoenfield, Oct 25 2019

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

A025639 Exponent of 2 (value of i) in n-th number of form 2^i*10^j.

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3, 13, 0, 10, 7, 4, 14, 1, 11, 8, 5, 15, 2, 12, 9, 6, 16, 3, 13, 0, 10, 7, 17, 4, 14, 1, 11, 8, 18, 5, 15, 2, 12, 9, 19, 6, 16, 3, 13, 0, 10, 20, 7, 17, 4, 14, 1, 11, 21, 8, 18, 5, 15, 2, 12, 22, 9, 19

Offset: 1

David W. Wilson

nonn

Cf. A025612.

A025684 Exponent of 10 (value of j) in n-th number of form 2^i*10^j.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 4, 1, 2, 3, 0, 4, 1, 2, 3, 0, 4, 1, 2, 3, 0, 4, 1, 5, 2, 3, 0, 4, 1, 5, 2, 3, 0, 4, 1, 5, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 4, 1, 5, 2, 6, 3, 0, 4, 1, 5, 2, 6, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 5

Offset: 1

David W. Wilson

nonn

Cf. A025612.

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

A036311 Composite numbers whose prime factors contain no digits other than 2 and 5.

Original entry on oeis.org

4, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000, 2048, 2500, 2560, 3125, 3200, 4000, 4096, 5000, 5120, 6250, 6400, 8000, 8192, 10000, 10240, 12500, 12800

Offset: 1

Patrick De Geest, Dec 15 1998

A003592 with 1, 2 and 5 removed. - Robert Israel, Apr 29 2018

Robert Israel, Table of n, a(n) for n = 1..10000 (first 131 terms from Vincenzo Librandi)
Index entries for sequences related to prime factors.

Cf. A003592, A036302-A036325, A025612.

[n: n in [4..13000] | not IsPrime(n) and forall{f: f in PrimeDivisors(n) | Intseq(f) subset [2,5]}]; // Bruno Berselli, Aug 26 2013

N:= 20000: # to get all terms <= N
S:= {seq(seq(2^i*5^j,i=0..ilog2(N/5^j)),j=0..floor(log[5](N)))} minus {1,2,5}:
sort(convert(S,list)); # Robert Israel, Apr 29 2018

dpfQ[n_]:=Module[{d=Union[Flatten[IntegerDigits/@Transpose[FactorInteger[n]][[1]]]]}, !PrimeQ[n]&&(d == {2}||d == {5}||d == {2, 5})]; Select[Range[15000], dpfQ] (* Vincenzo Librandi, Aug 25 2013 *)

Sum_{n>=1} 1/a(n) = 4/5. - Amiram Eldar, May 18 2022~

cp's OEIS Frontend

A032533 Numbers that, when expressed in base 2 and then interpreted in base 10, yield a multiple of the original number.

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

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A025639 Exponent of 2 (value of i) in n-th number of form 2^i*10^j.

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A025684 Exponent of 10 (value of j) in n-th number of form 2^i*10^j.

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

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Mathematica

Python

Python

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A036311 Composite numbers whose prime factors contain no digits other than 2 and 5.

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Magma

Maple

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