A055253 -id:A055253 - cp's OEIS frontend

A034887 Number of digits in 2^n.

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

Offset: 0

N. J. A. Sloane

The sequence consists of the positive integers, each appearing 3 or 4 times. - M. F. Hasler, Oct 08 2016

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Mersenne Number

Cf. A055253, A055254.
Cf. A000079, A055642.
See A125117 for the sequence of first differences.

[#Intseq(2^n): n in [0..100] ]; // Vincenzo Librandi, Jun 23 2015

seq(floor(n*ln(2)/ln(10))+1, n=0..100); # Jaap Spies, Dec 11 2003

Table[Length[IntegerDigits[2^n]], {n, 0, 100}] (* T. D. Noe, Feb 11 2013 *)
IntegerLength[2^Range[0,80]] (* Harvey P. Dale, Jul 28 2017 *)

A034887(n)=n*log(2)\log(10)+1 \\ or: { a(n)=#digits(1<M. F. Hasler, Oct 08 2016

def a(n): return len(str(1 << n))
print([a(n) for n in range(73)]) # Michael S. Branicky, Dec 23 2022

a(n) = floor(n*log_10(2)) + 1. E.g., a(10)=4 because 2^10 = 1024 and floor(10*log_10(2)) + 1 = 3 + 1 = 4. - Jaap Spies, Dec 11 2003

a(n) = A055642(2^n) = A055642(A000079(n)).

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

A272896 Difference between the number of odd and even digits in the decimal expansion of 2^n.

Original entry on oeis.org

1, -1, -1, -1, 0, 0, -2, -1, -1, 1, -2, -4, -2, 0, -1, -1, 1, 2, -4, -4, -1, 1, -1, -5, 2, 2, -4, 1, -3, 1, 0, -4, -2, 2, 3, 3, 1, 4, -2, 2, 5, 3, -1, -5, -2, -2, -2, 1, -1, 3, -4, 0, 2, 2, -1, -1, 5, 2, 2, -4, -3, 1, -5, -1, 0, 0, -6, 3, 5, 5, 2, -10, -8, 2, -3, 7, 9, 0, 0

Offset: 0

Seiichi Manyama, May 09 2016

All vanishing entries are a(A272898(k)) = 0, k >= 1. - Wolfdieter Lang, May 24 2016

			2^10 = 1024, 2^11 = 2048, 2^12 = 4096, 2^13 = 8192.
So a(10) = 1 - 3 = -2, a(11) = 0 - 4 = -4, a(12) = 1 - 3 = -2, a(13) = 2 - 2 = 0.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A000079, A055253, A055254, A196563, A196564, A272898.

Table[Count[#, ?OddQ] - Count[#, ?EvenQ] &@ IntegerDigits[2^n], {n, 0, 100}] (* Michael De Vlieger, May 09 2016 *)

a(n) = #select(x -> x%2, digits(2^n)) - #select(x -> !(x%2), digits(2^n));
for(n=0, 78, print1(a(n),", ")) \\ Indranil Ghosh, Mar 13 2017

def A272896(n):
    x=y=0
    for i in str(2**n):
        if int(i)%2: x+=1
        else: y+=1
    return x - y # Indranil Ghosh, Mar 13 2017

def a(n)
  str = (2 ** n).to_s
  str.size - str.split('').map(&:to_i).select{|i| i % 2 == 0}.size * 2
end
(0..n).each{|i| p a(i)}

a(n) = A055254(n) - A055253(n) = A196564(2^n) - A196563(2^n). - Indranil Ghosh, Mar 13 2017

A096614 Decimal expansion of Sum_{n>=1} f(2^n)/2^n, where f(n) is the number of even digits in n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jun 30 2004

This constant is transcendental. If the number of even digits is replaced with the number of odd digits, then the sum will be 1/9. (Borwein et al. 2004). - Amiram Eldar, Nov 14 2020

			1.03160638...

Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, 2004, pp. 14-15.

Eric Weisstein's World of Mathematics, Digit Count.
Index entries for transcendental numbers

Cf. A055253.

RealDigits[-1/9 + Sum[(1 + Floor[k*Log10[2]])/2^k, {k, 1, 350}], 10,
100][[1]] (* Amiram Eldar, Nov 14 2020 *)

-1/9 + suminf(k=1, (1 + floor(k * log(2)/log(10)))/2^k) \\ Michel Marcus, Nov 14 2020

Equals -1/9 + Sum_{k>=1} (1 + floor(k * log_10(2)))/2^k. - Amiram Eldar, Nov 14 2020

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A034887 Number of digits in 2^n.

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A055254 Number of odd digits in 2^n.

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A272896 Difference between the number of odd and even digits in the decimal expansion of 2^n.

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A096614 Decimal expansion of Sum_{n>=1} f(2^n)/2^n, where f(n) is the number of even digits in n.

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