A055253 -id:A055253

Search: a055253 -id:a055253

Displaying 1-4 of 4 results found.

page 1

A034887

Number of digits in 2^n.

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Mersenne Number

FORMULA

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)).

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

(Python)

def a(n): return len(str(1 << n))

print([a(n) for n in range(73)]) # Michael S. Branicky, Dec 23 2022

CROSSREFS

Cf. A055253, A055254.

Cf. A000079, A055642.

See A125117 for the sequence of first differences.

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A055254

Number of odd digits in 2^n.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

COMMENTS

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

REFERENCES

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

LINKS

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).

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(Perl) sub a{my $m; map $m+=1&$_, split //, 1<<pop; $m}

(Python)

def a(n): return sum(1 for d in str(1<<n) if d in "13579")

print([a(n) for n in range(91)]) # Michael S. Branicky, Dec 23 2022

CROSSREFS

Cf. A055253, A034887.

KEYWORD

nonn,base,easy

AUTHOR

Asher Auel, May 05 2000

EXTENSIONS

More terms from Jaap Spies, Dec 30 2003

STATUS

approved

A272896

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(Ruby)

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)}

(PARI) 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

(Python)

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

CROSSREFS

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

KEYWORD

sign,base

AUTHOR

Seiichi Manyama, May 09 2016

STATUS

approved

A096614

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..106.

Eric Weisstein's World of Mathematics, Digit Count.

Index entries for transcendental numbers

FORMULA

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

EXAMPLE

1.03160638...

MATHEMATICA

RealDigits[-1/9 + Sum[(1 + Floor[k*Log10[2]])/2^k, {k, 1, 350}], 10,

100][[1]] (* Amiram Eldar, Nov 14 2020 *)

PROG

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

CROSSREFS

Cf. A055253.

KEYWORD

nonn,cons,base

AUTHOR

Eric W. Weisstein, Jun 30 2004

STATUS

approved

page 1

Search completed in 0.008 seconds