cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-26 of 26 results.

A102674 Number of digits >= 4 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 17, 18, 18, 18, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 48, 50, 52, 54, 56, 57, 58, 59, 60, 62, 64, 66, 68, 70, 72, 73, 74, 75, 76, 78
Offset: 0

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

The total number of digits >= 4 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Links

Crossrefs

Partial sums of A102673.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

Programs

  • Maple
    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(add(p(i),i=0..n),n=0..90); # Emeric Deutsch, Feb 22 2005
  • Mathematica
    Accumulate[Table[Total[Drop[Most[DigitCount[n]],3]],{n,0,80}]] (* Harvey P. Dale, Nov 27 2015 *)

Formula

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 3/5)*(2n + 2 + (1/5 - floor(n/10^j + 3/5))*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)*A102673(n) + (1/2)*Sum_{j=1..m+1} (((1/5)*floor(n/10^j + 3/5) + floor(n/10^j))*10^j - (floor(n/10^j + 3/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m - 1) = 6*m*10^(m-1).
(This is the total number of digits >= 4 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(4*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

Extensions

More terms from Emeric Deutsch, Feb 22 2005

A102675 Number of digits >= 5 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

a(n) = 0 iff n is in A007091 (numbers in base 5). - Bernard Schott, Feb 02 2023

References

  • Curtis Cooper, Number of large digits in the positive integers not exceeding n, Abstracts Amer. Math. Soc., 25 (No. 1, 2004), p. 38, Abstract 993-11-964.

Links

Crossrefs

Cf. A007091 (indices of 0's), A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564, A102676 (partial sums).
Cf. A000120, A000788, A023416, A059015 (for base 2).

Programs

  • Maple
    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]>=5 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..120); # Emeric Deutsch, Feb 23 2005
  • Mathematica
    Table[Count[IntegerDigits[n],?(#>4&)],{n,0,120}] (* _Harvey P. Dale, Nov 13 2013 *)

Formula

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 1/2) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(5*10^j) - x^(10*10^j))/(1 - x^10^(j+1)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} x^(5*10^j)/(1 + x^(5*10^j)). (End)

Extensions

More terms from Emeric Deutsch, Feb 23 2005

A102676 Number of digits >= 5 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 10, 10, 10, 11, 12, 13, 14, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 20, 20, 20, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 56, 57, 58, 59, 60, 62
Offset: 0

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

The total number of digits >= 5 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

References

  • Curtis Cooper, Number of large digits in the positive integers not exceeding n, Abstracts Amer. Math. Soc., 25 (No. 1, 2004), p. 38, Abstract 993-11-964.

Links

Crossrefs

Partial sums of A102675.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

Programs

  • Maple
    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]>=5 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..83); # Emeric Deutsch, Feb 23 2005
  • Mathematica
    Accumulate[Table[Total[Take[DigitCount[n],{5,9}]],{n,0,80}]] (* Harvey P. Dale, Apr 27 2015 *)

Formula

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/2)*(2n + 2 - floor(n/10^j + 1/2)*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)*A102675(n) + (1/2)*Sum_{j=1..m+1} (floor(n/10^j)*10^j - (floor(n/10^j + 1/2)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 5*m*10^(m-1).
(This is the total number of digits >= 5 occurring in all the numbers with <= m places.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(5*10^j) - x^(10*10^j))/(1-x^10^(j+1)).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} x^(5*10^j)/(1+x^(5*10^j)). (End)

Extensions

More terms from Emeric Deutsch, Feb 23 2005

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    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

Formula

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)

Extensions

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

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Formula

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)

Extensions

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

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Maple
    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

Formula

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)

Extensions

More terms from Emeric Deutsch, Feb 23 2005
An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009
Previous Showing 21-26 of 26 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.