A196564 -id:A196564 - cp's OEIS frontend

A102678 Number of digits >= 6 in the decimal representations of all integers from 0 to n.

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 20, 20, 20, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 48

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 6 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Partial sums of A102677.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums of A102669.
Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=6 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..86); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 2/5)*(2n + 2 - (1/5 + floor(n/10^j + 2/5))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102677(n) + (1/2)*Sum_{j=1..m+1} ((-1/5*floor(n/10^j + 2/5) + floor(n/10^j))*10^j - (floor(n/10^j + 2/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 4*m*10^(m-1).
(this is total number of digits >= 6 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(6*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

A102680 Number of digits >= 7 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 7 occurring in all the numbers 0, 1, 2, ..., n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Partial sums of A102679.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums of A102669.
Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=7 then ct:=ct+1 else ct:=ct fi od: ct: end:
seq(add(p(i),i=0..n), n=0..90);
# Emeric Deutsch

Accumulate[Table[Count[IntegerDigits[n],?(#>6&)],{n,0,90}]] (* _Harvey P. Dale, Sep 04 2018 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 7/10)*(2n + 2 - (2/5 + floor(n/10^j + 7/10))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m=floor(log_10(n)).
a(n) = (n+1)*A102679(n) + (1/2)*Sum_{j=1..m+1} (((-2/5)*floor(n/10^j + 7/10) + floor(n/10^j))*10^j - (floor(n/10^j + 7/10)^2 - floor(n/10^j)^2)*10^j), where m=floor(log_10(n)).
a(10^m-1) = 3*m*10^(m-1).
(this is total number of digits >= 7 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(7*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A102682 Number of digits >= 8 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 8 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Partial sums of A102681.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=8 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..95); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/5)*(2n + 2 - (3/5 + floor(n/10^j + 1/5))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102681(n) + (1/2)*Sum_{j=1..m+1} ((-3/5*floor(n/10^j + 1/5) + floor(n/10^j))*10^j - (floor(n/10^j + 1/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 2*m*10^(m-1). (this is total number of digits >= 8 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(8*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

A136610 Number of odd digits in Fibonacci numbers.

Original entry on oeis.org

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

Offset: 0

Parthasarathy Nambi, May 11 2008

			1597 = Fibonacci(17) and has four odd digits, so a(17) = 4.

Ron Knott, The Fibonacci numbers.

Cf. A000045, A138468, A196564.

Cf. A060384, A085855. - R. J. Mathar, Jul 08 2009

nodss := proc(n) local dgs,d; dgs := convert(n,base,10) ; add( d mod 2, d=dgs) ; end: A136610 :=proc(n) nodss(combinat[fibonacci](n)) ; end: seq( A136610(n),n=0..80) ; # R. J. Mathar, Jul 08 2009

a[n_]:=Total[Boole[OddQ/@IntegerDigits[Fibonacci[n]]]] (* James C. McMahon, May 06 2025 *)

a(n) = A196564(A000045(n)). - Michel Marcus, May 06 2025

a(n) = A060384(n) - A138468(n). - James C. McMahon, Jun 07 2025

a(13) corrected and more terms added by R. J. Mathar, Jul 08 2009

A294601 Numbers with exactly one odd decimal digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 120, 122, 124, 126, 128, 140, 142, 144, 146, 148, 160, 162, 164, 166, 168, 180

Offset: 1

Robert Israel, Nov 03 2017

First differs from A054684 at position 56.
Numbers n such that A196564(n) = 1. - Felix Fröhlich, Nov 03 2017
There are (1+4*d)*5^(d-1) = 5*A081040(d+1) terms with d digits. - Robert Israel, Nov 06 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014263, A054684, A081040, A196564, A275775.

Res:= NULL:
for t from 0 to 1000 do
  if nops(select(type,convert(t,base,10),odd))=1 then Res:= Res,t fi
od:
Res;

Select[Range@ 200, Count[IntegerDigits@ #, ?OddQ] == 1 &] (* _Michael De Vlieger, Nov 03 2017 *)

a196564(n) = #select(x->x%2, digits(n)) \\ after Michel Marcus
is(n) = a196564(n)==1 \\ Felix Fröhlich, Nov 03 2017

A308005 A modified Sisyphus function: a(n) = concatenation of (number of odd digits in n) (number of digits in n) (number of even digits in n).

Original entry on oeis.org

11, 110, 11, 110, 11, 110, 11, 110, 11, 110, 121, 220, 121, 220, 121, 220, 121, 220, 121, 220, 22, 121, 22, 121, 22, 121, 22, 121, 22, 121, 121, 220, 121, 220, 121, 220, 121, 220, 121, 220, 22, 121, 22, 121, 22, 121, 22, 121, 22, 121, 121, 220, 121, 220, 121, 220, 121, 220, 121, 220, 22, 121, 22, 121, 22

Offset: 0

N. J. A. Sloane, May 12 2019

If we start with n and repeatedly apply the map i -> a(i), it appears that we eventually reach one of the two fixed points 22 or 231, or enter the two-cycle (33, 220). Are there any other possibilities? This is in contrast to the behavior of the closely related A308003.

			11 has 2 digits, both odd, so a(11)=220.
12 has 2 digits, one even and one odd, so a(12)=121. Then a(121) = 231, a fixed point.
22 has two digits, both even, so 22 -> 22, another fixed point  (leading zeros are omitted).

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A308003.

Maple code based on R. J. Mathar's code for A171797:
nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n, base, 10) do if type(d, 'even') then a :=a +1 ; end if; end do; a ; end proc:
cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:
catL := proc(L) local a, i; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:
A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:
A308005 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1-n2, n1, n2]) ; end proc:
[seq(A308005(n), n=0..80)];

A372585 Decimal expansion of (25/297)*Pi^2.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 06 2024

			0.8307747812364779982183914983060733278883585359630295140...

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.
Index entries for transcendental numbers

Cf. A000796, A002388, A196564, A372609.

Mathematica
```
First[RealDigits[25*Pi^2/297, 10, 100]]
```

Equals Sum_{k >= 1} A196564(k)*(1/k^2 - 1/(k+1)^2). See Sum 3 in Borwein and Borwein (1992), p. 622.

A372609 Decimal expansion of (10/9)*log(2).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 07 2024

			0.77016353395549478824136902384241840897277792706695028235631...

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.
Index entries for transcendental numbers

Cf. A002162, A193535, A196564, A372585.

Mathematica
```
First[RealDigits[10*Log[2]/9, 10, 100]]
```

log(2)*10/9 \\ Charles R Greathouse IV, May 18 2024

Equals Sum_{k >= 1} A196564(k)/(k*(k + 1)). See eq. 2.6 and Sum 13 in Borwein and Borwein (1992), p. 626.

Equals (10/3)*A193535. - Hugo Pfoertner, May 07 2024

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

A307153 Sequence gives pair of terms giving the numbers of previous even digits and previous odd digits; a(0)=0.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 14, 2, 15, 3, 18, 4, 19, 5, 22, 7, 23, 8, 24, 11, 26, 13, 28, 15, 30, 16, 32, 18, 34, 20, 35, 22, 37, 24, 39, 26, 41, 29, 43, 31, 46, 33, 48, 35, 50, 36, 52, 38, 54, 40, 55, 42, 57, 44, 59, 46, 61, 49, 63, 51, 66, 53, 68

Offset: 0

Paolo P. Lava, Mar 27 2019

Up to n = 10^5, any integer generally appears 0, 1 or 2 times. Only 248, 428 and 806 appear 3 times and 1 appears 8 times.
Are there any numbers that appear 4, 5 or more times?
From Giovanni Resta, Apr 01 2019: (Start)
4 times: 15711971, 22606282, 22826268, ...
5 times: 42862042, 44464482, 82802082, ...
6 times: 224026426, 224028040, 224042062, ...
7 times: 242620882, 244220442, 260088080, ...
Therefore, the first terms that appear n times, with n >= 0, are 6, 0, 3, 248, 15711971, 42862042, 224026426, 242620882, 1, ... (End)

			a(1) = 1 because there is only one even digit before a(1): a(0) = 0.
a(2) = 1 because there is only one odd digit before a(2): a(1) = 1. Etc.

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

Cf. A196563, A196564.

P:=proc(q) local a,b,d,d1,k,n,p,p1; a:=[0]: p:=1; d:=0;
for n from 2 to q do a:=[op(a),p]: b:=[op(convert(p,base,10))]:
p1:=0: d1:=0: for k from 1 to nops(b) do if b[k] mod 2=0
then p1:=p1+1: else d1:=d1+1: fi; od; d:=d+d1: p:=p+p1:
a:=[op(a),d]: b:=[op(convert(d,base,10))]: p1:=0: d1:=0:
for k from 1 to nops(b) do if b[k] mod 2=0 then p1:=p1+1:
else d1:=d1+1: fi; od; d:=d+d1: p:=p+p1: od; op(a); end: P(35);

nb = [0,0]; for (n=1, 71, print1 (v=nb[1+n%2]", "); apply(d -> nb[1+d%2]++, if (v, digits(v), [0]))) \\ Rémy Sigrist, May 04 2019

a(2n+1) = total number of even digits from a(0) to a(2n).

a(2n+2) = total number of odd digits from a(0) to a(2n+1).

cp's OEIS Frontend

A102678 Number of digits >= 6 in the decimal representations of all integers from 0 to n.

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A102680 Number of digits >= 7 in the decimal representations of all integers from 0 to n.

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A102682 Number of digits >= 8 in the decimal representations of all integers from 0 to n.

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A136610 Number of odd digits in Fibonacci numbers.

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A294601 Numbers with exactly one odd decimal digit.

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PARI

A308005 A modified Sisyphus function: a(n) = concatenation of (number of odd digits in n) (number of digits in n) (number of even digits in n).

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A372585 Decimal expansion of (25/297)*Pi^2.

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A372609 Decimal expansion of (10/9)*log(2).