A035014 -id:A035014 - cp's OEIS frontend

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

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.

A140288 The least n-digit multiple of 5^n using the decimal digits {1, 2, 3, 4, 5} exclusively.

Original entry on oeis.org

5, 25, 125, 3125, 53125, 453125, 4453125, 14453125, 314453125, 2314453125, 22314453125, 122314453125, 4122314453125, 44122314453125, 444122314453125, 4444122314453125, 54444122314453125, 254444122314453125, 1254444122314453125, 21254444122314453125

Offset: 1

Michel Criton (mcriton(AT)wanadoo.fr), May 24 2008

			a(5) = 53125 = 17 * 5^5.

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

Cf. A053312, A035014.

lista(nn) = {v = vector(nn); v[1] = 5; for(k = 2, nn, j = 1; while((j*10^(k-1) + v[k-1]) % 5^k > 0, j++); v[k] = j*10^(k-1) + v[k-1]); for(k = 1, nn, print1(v[k], ", "));} \\ Jinyuan Wang, Aug 27 2019

To obtain the (n+1)-th term, write the n-th term as k * 5^n. Multiply k by a multiple of 2^n to get a multiple of 5. Add the multiplicator of 2^n to the left of the n-th term.

More terms from Alois P. Heinz, Apr 05 2017

A050625 Divisible by 3^k (where k is digit length of a(n)).

Original entry on oeis.org

3, 6, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 567, 594, 621, 648, 675, 702, 729, 756, 783, 810, 837, 864, 891, 918, 945, 972, 999, 1053, 1134, 1215, 1296, 1377, 1458

Offset: 1

Patrick De Geest, Jun 15 1999

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A035014, A050622, A050623.

Flatten[Table[3^n*Range[Ceiling[10^(n-1)/3^n],Floor[(10^n-1)/3^n]],{n,4}]] (* Harvey P. Dale, Feb 26 2015 *)
Select[Range[1500],Mod[#,3^IntegerLength[#]]==0&] (* Harvey P. Dale, Apr 04 2020 *)

A053313 a(n) contains n digits (either '2' or '9') and is divisible by 2^n.

Original entry on oeis.org

2, 92, 992, 2992, 92992, 292992, 2292992, 22292992, 222292992, 2222292992, 22222292992, 922222292992, 9922222292992, 29922222292992, 929922222292992, 9929922222292992, 99929922222292992, 999929922222292992

Offset: 1

Henry Bottomley, Mar 06 2000

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

Cf. A023400, A050621, A050622, A035014.

Select[Flatten[Table[FromDigits/@Tuples[{2,9},n],{n,18}]],Divisible[ #,2^IntegerLength[ #]]&] (* Harvey P. Dale, Feb 07 2015 *)

a(n)=a(n-1)+10^(n-1)*(2+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 a 2, if not then n-th term begins with a 7.

A053314 a(n) contains n digits (either '1' or '4') and is divisible by 2^n.

Original entry on oeis.org

4, 44, 144, 4144, 14144, 414144, 1414144, 41414144, 441414144, 1441414144, 11441414144, 411441414144, 4411441414144, 44411441414144, 444411441414144, 1444411441414144, 41444411441414144, 441444411441414144

Offset: 1

Henry Bottomley, Mar 06 2000

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

Cf. A023401, A050621, A050622, A035014.

A[1]:= 4:
for n from 2 to 100 do
   if A[n-1] mod 2^n = 0 then A[n]:= A[n-1]+4*10^(n-1)
   else A[n]:= A[n-1]+10^(n-1)
fi
od:
seq(A[i],i=1..100); # Robert Israel, Oct 27 2019

nxt[{n_,a_}]:={n+1,If[Divisible[a,2^(n+1)],4*10^IntegerLength[a]+ a, 10^IntegerLength[ a]+a]}; NestList[nxt,{1,4},20][[All,2]] (* Harvey P. Dale, Oct 30 2022 *)

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 1.

Formula corrected by Robert Israel, Oct 27 2019

A053315 a(n) contains n digits (either '4' or '5') and is divisible by 2^n.

Original entry on oeis.org

4, 44, 544, 4544, 44544, 444544, 4444544, 54444544, 454444544, 5454444544, 45454444544, 545454444544, 5545454444544, 55545454444544, 555545454444544, 4555545454444544, 44555545454444544, 544555545454444544

Offset: 1

Henry Bottomley, Mar 06 2000

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

Cf. A023403, A050621, A050622, A035014.

A[1]:= 4:
for n from 2 to 100 do
   if A[n-1] mod 2^n = 0 then A[n]:= A[n-1]+4*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

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); 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 5.

Formula corrected by Robert Israel, Oct 27 2019

A053333 a(n) contains n digits (either '4' or '9') and is divisible by 2^n.

Original entry on oeis.org

4, 44, 944, 4944, 94944, 994944, 4994944, 94994944, 494994944, 9494994944, 49494994944, 449494994944, 9449494994944, 99449494994944, 499449494994944, 9499449494994944, 49499449494994944, 949499449494994944

Offset: 1

Henry Bottomley, Mar 06 2000

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

Cf. A023405, A050621, A050622, A035014.

nxt[{n_,a_}]:=Module[{fd=If[Mod[a,2^(n+1)]==0,4,9]},{n+1,fd 10^IntegerLength[a]+a}]; NestList[ nxt,{1,4},20][[;;,2]] (* Harvey P. Dale, Jul 14 2023 *)

a(n)=a(n-1)+10^(n-1)*(4+5*[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 9.

cp's OEIS Frontend

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.

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

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

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A140288 The least n-digit multiple of 5^n using the decimal digits {1, 2, 3, 4, 5} exclusively.

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A050625 Divisible by 3^k (where k is digit length of a(n)).

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A053313 a(n) contains n digits (either '2' or '9') and is divisible by 2^n.

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A053314 a(n) contains n digits (either '1' or '4') and is divisible by 2^n.

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A053315 a(n) contains n digits (either '4' or '5') and is divisible by 2^n.

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