A055254 -id:A055254

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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

A055253

Number of even digits in 2^n.

+10
5

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

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

OFFSET

0,7

LINKS

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(Python)

def a(n): return sum(1 for d in str(1<<n) if d in "02468")

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

CROSSREFS

Cf. A055254, 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

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