A034887 -id:A034887 - cp's OEIS frontend

A125117 First differences of A034887.

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jan 10 2007

This sequence is not periodic because log(2)/log(10) is an irrational number. - T. D. Noe, Jan 10 2007

The sequence consists only of 0's and 1's. Sequence A276397 (with a 0 prefixed) is similar but differs from a(42) on. Sequence A144597 differs only from a(102) on. - M. F. Hasler, Oct 07 2016

			a(1)=0 because 2^(1+1)=4 (one digit) 2^1=2 (one digit) and the difference is 0.
a(3)=1 because 2^(3+1)=16 (two digits) 2^(3)=8 (one digit) and the difference is 1.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A034887, A144597, A276397.

P:=proc(n) local i,j,k,w,old; k:=2; for i from 1 by 1 to n do j:=k^i; w:=0; while j>0 do w:=w+1; j:=trunc(j/10); od; if i>1 then print(w-old); old:=w; else old:=w; fi; od; end: P(1000);

Differences[IntegerLength[2^Range[0, 100]]] (* Paolo Xausa, Jun 08 2024 *)

a(n)=logint(2^(n+1),10)-logint(2^n,10) \\ Charles R Greathouse IV, Oct 19 2016

a(n) = number_of_digits{2^(n+1)} - number_of_digits{2^(n)} with n>=0.

A372804 Decimal expansion of Sum_{k >= 1, A034887(k) even} 1/2^k.

Original entry on oeis.org

1, 1, 1, 2, 1, 9, 5, 1, 2, 1, 9, 5, 1, 2, 1, 9, 5, 1, 2, 1, 9, 5, 1, 2, 1, 9, 5, 1, 2, 2, 0, 4, 9, 7, 3, 0, 9, 2, 2, 7, 9, 8, 3, 3, 1, 2, 3, 0, 3, 3, 4, 3, 4, 2, 6, 8, 9, 1, 3, 7, 9, 4, 3, 8, 2, 9, 8, 8, 2, 3, 5, 9, 1, 3, 9, 7, 7, 2, 6, 2, 4, 1, 7, 4, 7, 8, 6, 6, 4, 2

Offset: 0

Paolo Xausa, May 13 2024

			0.1112195121951219512195121951220497309227983312303343426891...

Paolo Xausa, Table of n, a(n) for n = 0..10000
J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.

Cf. A000079, A034887.

First[RealDigits[Sum[If[OddQ[IntegerLength[2^k]], 0, 1/2^k], {k, 350}], 10, 100]]

Approximately 114/1025, correct to 30 digits: see Example 4.2 (b) in Borwein and Borwein (1992), p. 634.

A000689 Final decimal digit of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6

Offset: 0

N. J. A. Sloane

These are the analogs of the powers of 2 in carryless arithmetic mod 10.
Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (also see the first comment under A001148). - K.V.Iyer, Mar 12 2010
This is also the decimal expansion of 227/1818. - Kritsada Moomuang, Dec 21 2021

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ...

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000079, A000855, A008952, A034887, A173635.

a000689 n = a000689_list !! n
a000689_list = 1 : cycle [2,4,8,6]  -- Reinhard Zumkeller, Sep 15 2011

[2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011

Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

for(n=0,80, if(n,{x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3,", ")},{print1("1, ")}))

[power_mod(2,n,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 03 2009

Periodic with period 4.
a(n) = 2^n mod 10.
a(n) = A002081(n) - A002081(n-1), for n > 0.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n > 3.
G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). (End)
For n >= 1, a(n) = 10 - (4x^3 + 47x - 27x^2)/3, where x = (n+3) mod 4 + 1.
For n >= 1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ).
G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012
a(n) = 5 + cos((n*Pi)/2) - 3*sin((n*Pi)/2) for n >= 1. - Kritsada Moomuang, Dec 21 2021

A210434 Number of digits in 4^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 41

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

Since log10(4) = A114493 ~ 0.60205 (= twice log10(2) = 0.30102999566...), the first 98 terms are equal to floor(n*3/5)+1. - M. F. Hasler, Mar 31 2025

			a(4) = 3 because 4^4 = 256, which has 3 digits.
a(5) = 4 because 4^5 = 1024, which has 4 digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000302, A034887, A034888, A210062, A210435, A210436.

[#Intseq(4^n): n in [0..68]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(4^n): seq(a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[4^n]], {n, 0, 68}] (* Bruno Berselli, Mar 22 2012 *)

apply( {A210434(n)=logint(4^n,10)+1}, [0..66]) \\ M. F. Hasler, Mar 31 2025

a(n)=log(4)*n\log(10)+1 \\ correct up to n ~ 10^precision, with default precision = 38. - M. F. Hasler, Mar 31 2025

from math import log
def A210434(n): return int(n*log(4,10))+1 if n<1e16 else "not enough precision" # M. F. Hasler, Mar 31 2025

a(n) = A055642(A000302(n)) = A055642(4^n) = floor(log_10(10*(4^n))). - Jonathan Vos Post, Mar 22 2012

A210435 Number of digits in 5^n.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

			a(4) = 3 because 5^4 = 625, which has 3 digits.
a(5) = 4 because 5^5 = 3125, which has 4 digits.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ana Rechtman, Octobre 2015, 3e Défi, Images des Mathématiques, CNRS, 2015 (in French).

Cf. A000351, A055642.

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), A210062 (b=7).

[#Intseq(5^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(5^n): seq(a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[5^n]], {n, 0, 67}] (* Bruno Berselli, Mar 22 2012 *)
IntegerLength[5^Range[0,70]] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = #Str(5^n); \\ Michel Marcus, Oct 27 2015

a(n) = A055642(A000351(n)) = A055642(5^n) = floor(log_10(10*(5^n))). [Jonathan Vos Post, Mar 22 2012]

a(n) + A034887(n) = n+1. - Michel Marcus, Oct 27 2015

A210436 Number of digits in 6^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 49, 50, 50, 51, 52, 53

Offset: 0

Luc Comeau-Montasse, Mar 21 2012

			a(4) = 4 because 6^4 = 1296, which has 4 digits.
a(5) = 4 because 6^5 = 7776, which has 4 digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000400, A034887, A034888, A210062, A210434, A210435.

[#Intseq(6^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

a:= n-> length(6^n): seq (a(n), n=0..100); # Alois P. Heinz, Mar 22 2012

Table[Length[IntegerDigits[6^n]], {n, 0, 99}] (* Alonso del Arte, Mar 22 2012 *)

a(n) = A055642(A000400(n)) = A055642(6^n) = floor(log_10(10*(6^n))). - Jonathan Vos Post, Mar 23 2012

A210062 Number of digits in 7^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57

Offset: 0

Luc Comeau-Montasse, Mar 16 2012

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000420, A055642.

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), this sequence (b=7).

[#Intseq(7^n): n in [0..67]]; // Bruno Berselli, Mar 22 2012

Table[Length[IntegerDigits[7^n]], {n, 0, 100}] (* T. D. Noe, Mar 20 2012 *)

a(n) = A055642(A000420(n)) = A055642(7^n) = floor(log_10(10*(7^n))). [Jonathan Vos Post, Mar 23 2012]

A055253 Number of even digits in 2^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 4, 3, 2, 3, 3, 2, 2, 5, 5, 4, 3, 4, 6, 3, 3, 6, 4, 6, 4, 5, 7, 6, 4, 4, 4, 5, 4, 7, 5, 4, 5, 7, 9, 8, 8, 8, 7, 8, 6, 10, 8, 7, 7, 9, 9, 6, 8, 8, 11, 11, 9, 12, 10, 10, 10, 13, 9, 8, 8, 10, 16, 15, 10, 13, 8, 7, 12, 12, 14, 13, 12, 15, 11, 12, 14, 10, 14, 16, 14, 16

Offset: 0

Asher Auel, May 05 2000

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A055254, A034887.

A055253 := proc(val) local i, j, k, n; n := 2^val; j := 0; k := floor(ln(n)/ln(10))+1; for i from 1 to k do if (n mod 10) mod 2 = 0 then j := j+1 fi; n := floor(n/10); od; RETURN(j); end: seq(A055253(n),n=0..110); # Jaap Spies, Dec 30 2003

Table[Length@ Select[IntegerDigits[2^n], EvenQ], {n, 0, 120}] (* or *)
Table[Total@ Pick[DigitCount[2^n], {0, 1, 0, 1, 0, 1, 0, 1, 0, 1}, 1], {n, 0, 120}] (* Michael De Vlieger, May 01 2016 *)
Count[IntegerDigits[#],?EvenQ]&/@(2^Range[0,100]) (* _Harvey P. Dale, Mar 25 2020 *)

a(n) = #select(x->(x % 2) == 0, digits(2^n)); \\ Michel Marcus, May 01 2016

def a(n): return sum(1 for d in str(1<Michael S. Branicky, Dec 23 2022

More terms from Jaap Spies, Dec 30 2003

A055254 Number of odd digits in 2^n.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 2, 2, 3, 4, 1, 1, 3, 4, 3, 1, 5, 5, 2, 5, 3, 5, 5, 3, 4, 6, 7, 7, 6, 8, 5, 7, 9, 8, 6, 4, 6, 6, 6, 8, 7, 9, 6, 8, 9, 9, 8, 8, 11, 10, 10, 7, 8, 10, 7, 9, 10, 10, 7, 12, 13, 13, 12, 6, 7, 12, 10, 15, 16, 12, 12, 10, 12, 13, 10, 14, 14, 12, 16, 13, 11, 13, 12

Offset: 0

Asher Auel, May 05 2000

Related sequence b(n) = Number of digits in 2^n that are at least 5. a(0) = 1, b(0) = 0 and a(n+1) = b(n), as a digit with value 5 of higher in 2^n will generate an odd digit in 2^(n+1). In the Nieuw Archief voor Wiskunde link there is a proof that sum(k>=, b(k)/2^k) = 2/9. - Jaap Spies, Mar 13 2009

			2^30 = 1073741824 and 1073741824 contains 5 odd decimal digits hence a(30)=5.

J. Borwein, D. Bailey and R. Girgensohn, Experimentation in mathematics : computational paths to discovery, A. K. Peters, 2004, pp. 14-15.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
D. Bowman and T. White, Proposers, Problem 6609, A rational sum, Amer. Math. Monthly, 98:3 (1991), 279-281.
Nieuw Archief voor Wiskunde, Problemen/UWC, p174-175, June 2004.
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).

Cf. A055253, A034887.

A055254 := proc(val) local i, j, k, n; n := 2^val; j := 0; k := floor(ln(n)/ln(10))+1; for i from 1 to k do if (n mod 10) mod 2 = 1 then j := j+1 fi; n := floor(n/10); od; RETURN(j); end: seq(A055254(n),n=0..110); # Jaap Spies

A055254[N_] := Count[ #, True] & /@ Map[OddQ, IntegerDigits /@ (2^# & /@ Range[N])] (* This generates a table of the number of odd digits in the first N powers of two *) (* Douglas Skinner (skinnerd(AT)comcast.net), Dec 06 2007 *)
Table[Count[IntegerDigits[2^n],?OddQ],{n,0,90}] (* _Harvey P. Dale, Mar 25 2015 *)

a(n)=my(d=digits(2^n)%2);sum(i=1,#d,d[i]) \\ Charles R Greathouse IV, Jun 04 2013

sub a{my $m;map $m+=1&$_,split //,1<

def a(n): return sum(1 for d in str(1<Michael S. Branicky, Dec 23 2022

Sum(k>=0,a(k)/2^k)=11/9 (for a proof see the comment above). [Corrected by Jaap Spies, Mar 13 2009]

More terms from Jaap Spies, Dec 30 2003

A077352 a(n) = (concatenation in ascending order of divisors of 2^n)/2^n.

Original entry on oeis.org

1, 6, 31, 156, 7801, 390051, 19502551, 9751275501, 4875637750501, 2437818875250501, 12189094376252505001, 60945471881262525005001, 304727359406312625025005001, 1523636797031563125125025005001, 76181839851578156256251250250050001, 3809091992578907812812562512502500050001

Offset: 0

Amarnath Murthy, Nov 05 2002

			a(6) = 1248163264/64 = 19502551.

Paolo Xausa, Table of n, a(n) for n = 0..80

Cf. A000079, A034887, A059268, A077351, A069872, A077353.

a:= n-> parse(cat(2^i$i=0..n))/2^n:
seq(a(n), n=0..15);  # Alois P. Heinz, May 16 2025

A077352[n_] := FromDigits[Flatten[IntegerDigits[Divisors[#]]]]/# & [2^n];
Array[A077352, 16, 0] (* or *)
FoldList[10^IntegerLength[2^#2]*#/2 + 1 &, 1, Range[15]] (* Paolo Xausa, May 19 2025 *)

For n>=1, a(n) = (a(n-1)*2^(n-1)*10^(floor(log_10(2^n))+1)+2^n)/2^n. - Sam Alexander, Feb 27 2005

Offset corrected by Sean A. Irvine, May 16 2025

cp's OEIS Frontend

A125117 First differences of A034887.

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A372804 Decimal expansion of Sum_{k >= 1, A034887(k) even} 1/2^k.

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A000689 Final decimal digit of 2^n.

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A210434 Number of digits in 4^n.

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A210435 Number of digits in 5^n.

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A210436 Number of digits in 6^n.

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