A045507 -id:A045507 - cp's OEIS frontend

A067097 Floor[X/Y] where X = concatenation in increasing order of first n powers of 2 and Y = that of first n natural numbers.

2, 2, 2, 20, 201, 2010, 201012, 20101226, 2010122457, 201012245610, 20101224560848, 2010122456084687, 201012245608468521, 201012245608468519453, 201012245608468519428723, 201012245608468519428463029, 2010122456084685194284602619644

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(6)= floor [ 248163264/123456] = floor[2010.13530326594090202177293] = 2010.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095, A067096.

floor[A045507(n)/A019519(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2^k]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]
Table[Floor[FromDigits[Flatten[IntegerDigits/@(2^Range[n])]]/FromDigits[ Flatten[IntegerDigits/@Range[n]]]],{n,20}] (* Harvey P. Dale, Dec 30 2018 *)

More terms from Robert G. Wilson v, Jan 09 2002

More terms from Harvey P. Dale, Dec 30 2018

A051639 Concatenation of 3^k, k = 0,..,n.

Original entry on oeis.org

1, 13, 139, 13927, 1392781, 1392781243, 1392781243729, 13927812437292187, 139278124372921876561, 13927812437292187656119683, 1392781243729218765611968359049, 1392781243729218765611968359049177147, 1392781243729218765611968359049177147531441

Offset: 0

Felice Russo, Nov 15 1999

			139 belongs to the sequence because it is the concatenation of 3^0, 3^1 and 3^2.

A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000

Alois P. Heinz, Table of n, a(n) for n = 0..63

Cf. A000244. - R. J. Mathar, Oct 10 2010

Cf. A045507.

From R. J. Mathar, Oct 10 2010: (Start)
cat2 := proc(a,b) dgsb := max(1,ilog10(b)+1) ; a*10^dgsb+b ; end proc:
catL := proc(L) local a; a := op(1,L) ; for i from 2 to nops(L) do a := cat2(a,op(i,L)) ; end do; a; end proc:
A051639 := proc(n) catL([seq(3^k,k=0..n)]) ; end proc: seq(A051639(n),n=0..20) ; (End)
# second Maple program:
a:= proc(n) a(n):= `if`(n<0, 0, parse(cat(a(n-1), 3^n))) end:
seq(a(n), n=0..12);  # Alois P. Heinz, May 30 2021

With[{p3=3^Range[0,15]},Table[FromDigits[Flatten[IntegerDigits/@ Take[ p3,n]]],{n,15}]] (* Harvey P. Dale, Sep 13 2011 *)

Terms n>=7 corrected by R. J. Mathar, Oct 10 2010

A381259 Numbers obtained by concatenating powers of 2, sorted into increasing order.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 16, 18, 21, 22, 24, 28, 32, 41, 42, 44, 48, 64, 81, 82, 84, 88, 111, 112, 114, 116, 118, 121, 122, 124, 128, 132, 141, 142, 144, 148, 161, 162, 164, 168, 181, 182, 184, 188, 211, 212, 214, 216, 218, 221, 222, 224, 228, 232, 241, 242, 244, 248, 256, 264

Offset: 1

Stefano Spezia, Feb 18 2025

Take the list {2^i: i >= 0} and concatenate its terms (allowing multiple copies) in any order; then sort the result into increasing order.

The term a(32) = 128 is a power of 2 as well as the concatenation of several powers of 2. - Rémy Sigrist, Feb 20 2025

			11 is a term because it is the concatenation of 1 = 2^0 with itself;
12 is a term because it is the concatenation of 1 = 2^0 with 2 = 2^1;
32 is a term because it is equal to 2^5;
168 is a term because it is the concatenation of 16 = 2^4 with 8 = 2^3.
0 is not a term because it is not a power of 2.

Rémy Sigrist, Table of n, a(n) for n = 1..11921
Rémy Sigrist, PARI program

Supersequence of A028846.
Some subsequences: A000079, A045507, A178664.
Cf. A152242.

PARI
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A386552 Concatenate powers of 10.

Original entry on oeis.org

1, 110, 110100, 1101001000, 110100100010000, 110100100010000100000, 1101001000100001000001000000, 110100100010000100000100000010000000, 110100100010000100000100000010000000100000000, 1101001000100001000001000000100000001000000001000000000

Offset: 0

Jason Bard, Jul 25 2025

Binary version of A045507. Base-2 representation of A164894.

Concatenate first A000217(n+1) terms of A010054.

Jason Bard, Table of n, a(n) for n = 0..43

Cf. A000096, A000217, A010054, A011557, A045507, A051639, A101305, A133082, A164894, A242646.

a:= proc(n) option remember;
      `if`(n<0, 0, parse(cat(a(n-1), 10^n)))
    end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 28 2025

a[0] = 1; a[n_] := a[n - 1]*10^(n+1) + 10^n; Table[a[n], {n, 0, 9}]

def A386552(n): return 10**n*sum(10**(k*((n<<1)-k+1)>>1) for k in range(n+1)) # Chai Wah Wu, Aug 05 2025

a(n) = Sum_{k=1..n+1} 10^A133082(k,n+2).
a(n) = A101305(n) + 10^A000096(n).
For n >= 1, a(n) = 10^(n+1)*a(n-1)+10^n.
Number of digits in a(n) is A000217(n+1).

cp's OEIS Frontend

A067097 Floor[X/Y] where X = concatenation in increasing order of first n powers of 2 and Y = that of first n natural numbers.

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A051639 Concatenation of 3^k, k = 0,..,n.

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A381259 Numbers obtained by concatenating powers of 2, sorted into increasing order.

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A386552 Concatenate powers of 10.

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