A050622 -id:A050622 - cp's OEIS frontend

A035014 a(n) contains n digits (either '3' or '4') and is divisible by 2^n.

4, 44, 344, 3344, 33344, 433344, 3433344, 33433344, 333433344, 3333433344, 43333433344, 343333433344, 3343333433344, 33343333433344, 433343333433344, 3433343333433344, 43433343333433344, 443433343333433344, 3443433343333433344, 43443433343333433344

Offset: 1

J. Lowell

If (n-1)st term is divisible by 2^n, then n-th term begins with a 4. If not, then n-th term begins with a 3.

Proof of conjecture that a(n) ends with a(n-1): If a(n) is divisible by 2^n, then a(n) is divisible by 2^(n-1), so a(n)-k*10^(n-1) is divisible by 2^(n-1) for integer k, but if k is first digit of a(n) then a(n)-k*10^(n-1) is an (n-1)-digit number made up of 3s and 4s and divisible by 2^(n-1) and so must be a(n-1). - Henry Bottomley, Feb 14 2000

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 100 terms from Jon E. Schoenfield)

Cf. A050620, A050621, A050622, A023402.

A035014 := proc(n)
    option remember ;
      local pre;
      if n = 1 then
        4;
    else
        pre := procname(n-1) ;
        pre+10^(n-1)*(4-modp(pre/2^(n-1),2)) ;
    end if;
end proc: # R. J. Mathar, May 02 2014

a(n) = if (n==1, 4, a(n-1) + 10^(n-1)*(4-(a(n-1)/2^(n-1) % 2))); \\ Michel Marcus, Apr 07 2017

a(n) = a(n-1) + 10^(n-1)*(4-[a(n-1)/2^(n-1) mod 2]), i.e., a(n) ends with a(n-1). - Henry Bottomley, Feb 14 2000

Corrected and extended by Patrick De Geest, Jun 15 1999

More terms from Henry Bottomley, Feb 14 2000

A050621 Smallest n-digit number divisible by 2^n.

Original entry on oeis.org

2, 12, 104, 1008, 10016, 100032, 1000064, 10000128, 100000256, 1000000512, 10000001024, 100000002048, 1000000004096, 10000000008192, 100000000016384, 1000000000032768, 10000000000065536, 100000000000131072

Offset: 1

Patrick De Geest, Jun 15 1999

Quotients arising from this sequence give A034478 ((5^(n-1)+1)/2).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (12,-20).

Cf. A035014, A034478, A050622.

[2^(n-1)+10^(n-1): n in [1..21]]; // Vincenzo Librandi, Sep 12 2014

CoefficientList[Series[2 (1 - 6 x)/((1 - 2 x) (1 - 10 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 12 2014 *)

a(n) = 10^(n-1) + 2^(n-1) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = 10^(n-1) + 2^(n-1).
G.f.: Q(0) where Q(k)= 1 + 5^k/(1 - 2*x/(2*x + 5^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
G.f.: 2*x*(1-6*x)/((1-2*x)*(1-10*x)). - Vincenzo Librandi, Sep 12 2014
a(n) = 12*a(n-1) - 20*a(n-2) for n>1. - Vincenzo Librandi, Sep 12 2014

A053312 a(n) contains n digits (either '1' or '2') and is divisible by 2^n.

Original entry on oeis.org

2, 12, 112, 2112, 22112, 122112, 2122112, 12122112, 212122112, 1212122112, 11212122112, 111212122112, 1111212122112, 11111212122112, 211111212122112, 1211111212122112, 11211111212122112, 111211111212122112, 2111211111212122112, 12111211111212122112

Offset: 1

Henry Bottomley, Mar 06 2000

Corresponding quotients a(n) / 2^n are in A126933. - Bernard Schott, Mar 15 2023

			a(5) = 22112 since 22112 = 2^5 * 691 and 22112 contains 5 digits.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000079, A023396, A050621, A050622, A035014, A126933, A207778.

a:= proc(n) option remember; `if`(n=1, 2, (t-> parse(
      cat(`if`(irem(t, 2^n)=0, 2, 1), t)))(a(n-1)))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 02 2025

Select[Flatten[Table[FromDigits/@Tuples[{1,2},n],{n,20}]],Divisible[ #,2^IntegerLength[#]]&] (* Harvey P. Dale, Jul 01 2019 *)
RecurrenceTable[{a[1]==2, a[n]==a[n-1]+10^(n-1)(2-Mod[a[n-1]/2^(n-1),2])}, a[n], {n, 1, 20}] (* Paul C Abbott, Feb 03 2025 *)

from itertools import count, islice
def A053312_gen(): # generator of terms
    a = 0
    for n in count(0):
        yield (a:=a+(10**n if (a>>n)&1 else 10**n<<1))
A053312_list = list(islice(A053312_gen(),20)) # Chai Wah Wu, Mar 15 2023

a(n) = a(n-1) + 10^(n-1)*(2-[a(n-1)/2^(n-1) mod 2]), i.e., a(n) ends with a(n-1); if the (n-1)-th term is divisible by 2^n then the n-th term begins with a 2; if not, then the n-th term begins with a 1.

A053318 a(n) contains n digits (either '2' or '7') and is divisible by 2^n.

Original entry on oeis.org

2, 72, 272, 2272, 22272, 222272, 7222272, 27222272, 727222272, 2727222272, 72727222272, 772727222272, 7772727222272, 77772727222272, 277772727222272, 2277772727222272, 72277772727222272, 272277772727222272

Offset: 1

Henry Bottomley, Mar 06 2000

Robert Israel, Table of n, a(n) for n = 1..999

Cf. A023399, A050621, A050622, A035014.

A[1]:= 2:
for n from 2 to 100 do
   if A[n-1] mod 2^n = 0 then A[n]:= A[n-1]+2*10^(n-1)
   else A[n]:= A[n-1]+7*10^(n-1)
     fi
od:
seq(A[i],i=1..100); # Robert Israel, Oct 27 2019

Table[Select[FromDigits/@Tuples[{2,7},n],Mod[#,2^n]==0&],{n,18}]//Flatten (* Harvey P. Dale, Jul 14 2025 *)

a(n) = a(n-1) + 10^(n-1)*(2 + 5*(a(n-1)/2^(n-1) mod 2)), i.e., a(n) ends with a(n-1); if a(n-1) is divisible by 2^n then a(n) begins with a 2, if not then a(n) begins with a 7.

Formula corrected by Robert Israel, Oct 27 2019

A053316 a(n) contains n digits (either '2' or '3') and is divisible by 2^n.

Original entry on oeis.org

2, 32, 232, 3232, 23232, 223232, 2223232, 32223232, 232223232, 3232223232, 23232223232, 323232223232, 3323232223232, 23323232223232, 323323232223232, 3323323232223232, 33323323232223232, 333323323232223232

Offset: 1

Henry Bottomley, Mar 06 2000

Robert Israel, Table of n, a(n) for n = 1..999

Cf. A023397, A050621, A050622, A035014.

A[1]:= 2:
for n from 2 to 100 do
   if A[n-1] mod 2^n = 0 then A[n]:= A[n-1]+2*10^(n-1)
   else A[n]:= A[n-1]+3*10^(n-1)
     fi
od:
seq(A[i],i=1..100); # Robert Israel, Oct 27 2019

a(n) = a(n-1) + 10^(n-1)*(2 + (a(n-1)/2^(n-1) mod 2)), i.e., a(n) ends with a(n-1); if a(n-1) is divisible by 2^n then a(n) begins with a 2, if not then a(n) begins with a 3.

Formula corrected by Robert Israel, Oct 27 2019

A053317 a(n) contains n digits (either '2' or '5') and is divisible by 2^n.

Original entry on oeis.org

2, 52, 552, 5552, 55552, 255552, 5255552, 55255552, 255255552, 2255255552, 22255255552, 222255255552, 5222255255552, 55222255255552, 255222255255552, 2255222255255552, 22255222255255552, 222255222255255552

Offset: 1

Henry Bottomley, Mar 06 2000

Robert Israel, Table of n, a(n) for n = 1..999

Cf. A023398, A050621, A050622, A035014, A055880.

A[1]:= 2:
for n from 2 to 100 do
   if A[n-1] mod 2^n = 0 then A[n]:= A[n-1]+2*10^(n-1)
   else A[n]:= A[n-1]+5*10^(n-1)
fi
od:
seq(A[i],i=1..100); # Robert Israel, Oct 27 2019

Table[Select[FromDigits/@Tuples[{2,5},n],Divisible[#,2^n]&],{n,18}]//Flatten (* Harvey P. Dale, Oct 12 2022 *)

a(n) = a(n-1) + 10^(n-1)*(2 + 3*(a(n-1)/2^(n-1) mod 2)), i.e., a(n) ends with a(n-1); if a(n-1) is divisible by 2^n then a(n) begins with a 2, if not then a(n) begins with a 5.

Formula corrected by Robert Israel, Oct 27 2019

A053332 a(n) contains n digits (either '4' or '7') and is divisible by 2^n.

Original entry on oeis.org

4, 44, 744, 7744, 47744, 447744, 4447744, 44447744, 444447744, 4444447744, 74444447744, 474444447744, 4474444447744, 44474444447744, 444474444447744, 7444474444447744, 77444474444447744, 477444474444447744

Offset: 1

Henry Bottomley, Mar 06 2000

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A023404, A050621, A050622, A035014.

Flatten[Table[Select[FromDigits/@Tuples[{4,7},n],Divisible[#,2^n]&], {n,20}]] (* Harvey P. Dale, Jul 25 2011 *)

a(n) = a(n-1)+10^(n-1)*(4+3*[a(n-1)/2^(n-1) mod 2]) i.e. a(n) ends with a(n-1); if (n-1)-th term is divisible by 2^n then n-th term begins with a 4, if not then n-th term begins with a 7.

A053338 a(n) contains n digits (either '6' or '9') and is divisible by 2^n.

Original entry on oeis.org

6, 96, 696, 9696, 69696, 669696, 6669696, 96669696, 696669696, 9696669696, 69696669696, 969696669696, 9969696669696, 69969696669696, 969969696669696, 9969969696669696, 99969969696669696, 999969969696669696

Offset: 1

Henry Bottomley, Mar 06 2000

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A023410, A050621, A050622, A035014.

Table[Select[FromDigits/@Tuples[{6,9},n],Mod[#,2^n]==0&],{n,20}]//Flatten (* Harvey P. Dale, Sep 15 2023 *)

a(n)=a(n-1)+10^(n-1)*(6+3*[a(n-1)/2^(n-1) mod 2]) i.e. a(n) ends with a(n-1); if (n-1)-th term is divisible by 2^n then n-th term begins with a 6, if not then n-th term begins with a 9.

A053376 a(n) contains n digits (either '1' or '8') and is divisible by 2^n.

Original entry on oeis.org

8, 88, 888, 1888, 81888, 181888, 8181888, 18181888, 118181888, 8118181888, 88118181888, 888118181888, 8888118181888, 88888118181888, 888888118181888, 1888888118181888, 81888888118181888, 181888888118181888

Offset: 1

Henry Bottomley, Mar 06 2000

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A023411, A050621, A050622, A035014.

Table[Select[FromDigits/@Tuples[{1,8},n],Divisible[#,2^n]&],{n,20}]// Flatten (* Harvey P. Dale, Aug 20 2017 *)

a(n)=a(n-1)+10^(n-1)*(8-7*[a(n-1)/2^(n-1) mod 2]) i.e. a(n) ends with a(n-1); if (n-1)-th term is divisible by 2^n then n-th term begins with an 8, if not then n-th term begins with a 1.

A053380 a(n) contains n digits (either '8' or '9') and is divisible by 2^n.

Original entry on oeis.org

8, 88, 888, 9888, 89888, 989888, 9989888, 89989888, 989989888, 8989989888, 98989989888, 898989989888, 8898989989888, 98898989989888, 998898989989888, 8998898989989888, 98998898989989888, 898998898989989888

Offset: 1

Henry Bottomley, Mar 06 2000

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A023415, A050621, A050622, A035014.

a(n)=a(n-1)+10^(n-1)*(8+[a(n-1)/2^(n-1) mod 2]) i.e. a(n) ends with a(n-1); if (n-1)-th term is divisible by 2^n then n-th term begins with an 8, if not then n-th term begins with a 9.

cp's OEIS Frontend

A035014 a(n) contains n digits (either '3' or '4') and is divisible by 2^n.

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A050621 Smallest n-digit number divisible by 2^n.

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A053312 a(n) contains n digits (either '1' or '2') and is divisible by 2^n.

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A053318 a(n) contains n digits (either '2' or '7') and is divisible by 2^n.

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A053316 a(n) contains n digits (either '2' or '3') and is divisible by 2^n.

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A053317 a(n) contains n digits (either '2' or '5') and is divisible by 2^n.

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A053332 a(n) contains n digits (either '4' or '7') and is divisible by 2^n.

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A053338 a(n) contains n digits (either '6' or '9') and is divisible by 2^n.