A054899 -id:A054899 - cp's OEIS frontend

A102677 Number of digits >= 6 in decimal representation of n.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007092 (numbers in base 6). - Bernard Schott, Feb 02 2023

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

Cf. A007092, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102678.

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(p(n),n=0..116); # Emeric Deutsch, Feb 23 2005

Table[Total@ Take[Most@ DigitCount@ n, -4], {n, 0, 104}] (* Michael De Vlieger, Aug 17 2017 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 2/5) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*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

A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

Original entry on oeis.org

0, 1, 6, 7, 10, 11, 60, 61, 180, 181, 285, 300, 301, 575, 687, 754, 826, 827, 882, 883, 900, 901, 910, 911, 1254, 1305, 1311, 1326, 1327, 1335, 1377, 1383, 1386, 1387, 1395, 1431, 1506, 1507, 1532, 1626, 1627, 1650, 1651, 1890, 1891, 1955, 2013, 2036, 2040

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=6, since ds_2(6)=ds_3(6)=ds_5(6), where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0,3000],Length[Union[Total/@IntegerDigits[#,{2,3,5}]]]==1&] (* Harvey P. Dale, Sep 04 2014 *)

Added 0, Stanislav Sykora, May 06 2012

A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

Original entry on oeis.org

0, 1, 882, 883, 1386, 1387, 2502, 2503, 3453, 7555, 7652, 7665, 7931, 9751, 10101, 12250, 12251, 16893, 17010, 17011, 17515, 17550, 17551, 18285, 20301, 22050, 22051, 24406, 24407, 25053, 27503, 31654, 40930, 40931, 41951, 50878, 50879

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=882, since ds_2(882 )=ds_3(882 )=ds_5(882 )=ds_7(882 )=6, where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0, 32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 27 2016 *)
Select[Range[0,51000],Length[Union[Total/@IntegerDigits[#,{2,3,5,7}]]] == 1&] (* Harvey P. Dale, Sep 18 2019 *)

Added 0, Stanislav Sykora, May 06 2012

A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 03 2005

This is the total number of digits = 9 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 A102683.
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]>=9 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..105); # Emeric Deutsch, Feb 23 2005

Accumulate[DigitCount[Range[0,100],10,9]] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = sum(k=0, n, #select(x->(x==9), digits(k))); \\ Michel Marcus, Oct 03 2023

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/10)*(2n + 2 - (4/5 + floor(n/10^j + 1/10))*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)*A102683(n) + (1/2)*Sum_{j=1..m+1} ((-4/5*floor(n/10^j + 1/10) + floor(n/10^j))*10^j - (floor(n/10^j + 1/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = m*10^(m-1).
(this is total number of digits = 9 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(9*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

Definition revised by N. J. A. Sloane, Mar 30 2018

A054900 a(n) = Sum_{j >= 1} floor(n/16^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Henry Bottomley, May 23 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.

Cf. A053836, A054897, A054899, A067080, A098844, A132032.

m:=16;
function a(n) // a = A054900, m = 16
  if n eq 0 then return 0;
  else return a(Floor(n/m)) + Floor(n/m);
  end if; end function;
[a(n): n in [0..127]]; // G. C. Greubel, Apr 28 2023

a[n_, m_]:= If[n==0, 0, a[Floor[n/m], m] +Floor[n/m]];
Table[a[n, 16], {n,0,127}] (* G. C. Greubel, Apr 28 2023 *)

m=16 # a = A054900
def a(n): return 0 if (n==0) else a(n//m) + (n//m)
[a(n) for n in range(128)] # G. C. Greubel, Apr 28 2023

a(n) = (n - A053836(n))/15.
From Hieronymus Fischer, Aug 14 2007: (Start)
Recurrence:
a(n) = a(floor(n/16)) + floor(n/16).
a(16*n) = a(n) + n.
a(n*16^m) = a(n) + n*(16^m - 1)/15.
a(k*16^m) = k*(16^m - 1)/15, for 0 <= k < 16, m>=0.
Asymptotic behavior:
a(n) = n/15 + O(log(n)).
a(n+1) - a(n) = O(log(n)) (this follows from the inequalities below).
a(n) <= (n-1)/15; equality holds for powers of 16.
a(n) >= (n-15)/15 - floor(log_16(n)); equality holds for n = 16^m - 1, m > 0.
Limits:
lim inf (n/15 - a(n)) = 1/15, for n --> oo.
lim sup (n/15 - log_16(n) - a(n)) = 0, for n --> oo.
lim sup (a(n+1) - a(n) - log_16(n)) = 0, for n --> oo.
Series:
G.f.: (1/(1-x))*Sum_{k > 0} x^(16^k)/(1-x^(16^k)). (End)

A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

Original entry on oeis.org

1, 21, 261, 273, 17748, 17749, 20820, 20821, 65620, 65621, 70740, 70741, 83268, 83269, 86292, 86293, 1066068, 1066069, 1070420, 1135701, 1135893, 1135953, 5326161, 5330001, 5330241, 5330260, 5330261, 5506389, 5525829, 5526801, 5571909, 5574933, 5592321

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3, where ds_x=digital sum base x.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[5600000],Length[Union[Table[Total[IntegerDigits[#,n]],{n,2,4}]]]==1&] (* Harvey P. Dale, Aug 14 2013 *)

isok(n) = my(sd2=sumdigits(n, 2)); (sd2==sumdigits(n, 3)) && (sd2==sumdigits(n, 4)); \\ Michel Marcus, Aug 08 2018

a(31)-a(33) from Giovanni Resta, Aug 06 2018

A102670 Number of digits >= 2 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 53, 54, 56, 58, 60, 62, 64, 66, 68, 70, 71, 72, 74, 76, 78, 80, 82, 84, 86, 88, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 107, 108

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 2 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 A102669.
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]>=2 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..77); # Emeric Deutsch, Feb 23 2005

Accumulate[Table[Count[IntegerDigits[n],?(#>1&)],{n,0,80}]] (* _Harvey P. Dale, Apr 17 2014 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 0.8)*(2n + 2 + ((3/5) - floor(n/10^j + 4/5))*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)* A102669(n) + (1/2)*Sum_{j=1..m+1} (((3/5)*floor(n/10^j + 4/5) + floor(n/10^j))*10^j - (floor(n/10^j + 4/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m - 1) = 8*m*10^(m-1).
(This is the total number of digits >= 2 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(2*10^j) - x^(10*10^j))/(1 - x^10^(j+1)).
General formulas for the total number of digits >= d in the decimal representations of all integers from 0 to n.
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + (10-d)/10) *(2n + 2 + ((5-d)/5 - floor(n/10^j + (10-d)/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)*F(n,d) + (1/2)*Sum_{j=1..m+1} ((((5-d)/5)*floor(n/10^j + (10-d)/10) + floor(n/10^j))*10^j - (floor(n/10^j + (10-d)/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)) and F(n,d) = number of digits >= d in the decimal representation of n.
a(10^m - 1) = (10-d)*m*10^(m-1).
(This is the total number of digits >= d occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(d*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A102671 Number of digits >= 3 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 1, 1

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007089 (numbers in base 3). - Bernard Schott, Nov 20 2022

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

Cf. A007089, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102672.

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]>=3 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..116); # Emeric Deutsch, Feb 23 2005

Table[Count[IntegerDigits[n],?(#>2&)],{n,0,110}] (* _Harvey P. Dale, Mar 07 2012 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor((n/10^j) + 7/10) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(3*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A102672 Number of digits >= 3 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 32, 34, 36, 38, 39, 40, 41, 43, 45, 47, 49, 51, 53, 55, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 92, 94

Offset: 0

N. J. A. Sloane, Feb 03 2005

The total number of digits >= 3 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 A102671.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums of A102671.
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]>=3 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..80); # Emeric Deutsch, Feb 23 2005

Accumulate[Table[Count[IntegerDigits[n],?(#>2&)],{n,0,80}]] (* _Harvey P. Dale, Nov 23 2014 *)

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)*A102671(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) = 7*m*10^(m-1).
(This is the total number of digits >= 3 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(3*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A102673 Number of digits >= 4 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007090 (numbers in base 4). - Bernard Schott, Feb 01 2023

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

Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102674.

Cf. A000120, A000788, A007090, 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]>=4 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..125); # Emeric Deutsch, Feb 22 2005

Table[Total@ Take[DigitCount@ n, {4, 9}], {n, 0, 104}] (* Michael De Vlieger, Aug 17 2017 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 3/5) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(4*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 22 2005

cp's OEIS Frontend

A102677 Number of digits >= 6 in decimal representation of n.

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A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

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A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

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A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.

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A054900 a(n) = Sum_{j >= 1} floor(n/16^j).

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A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

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A102670 Number of digits >= 2 in the decimal representations of all integers from 0 to n.

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