A055641 -id:A055641 - cp's OEIS frontend

A316867 Number of times 6 appears in decimal expansion of n.

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

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 6.
a(6) = 1 since 6 appears once in the decimal expansion of 6.

Cf. A043513, A043514, A043515, A043516.

Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 6] &, 105, 0]
```

a(n) = #select(x->x==6, digits(n)); \\ Michel Marcus, Jul 20 2018

A061217 Number of zeros in the concatenation n(n-1)(n-2)(n-3)...321.

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, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 12, 13, 14

Offset: 1

Amarnath Murthy, Apr 22 2001

The number of zeros necessary to write down all the numbers 1, 2, ..., n. Thus, the partial sums of A055641 are given by a(n)+1 (for n>=1). - Hieronymus Fischer, Jun 12 2012

			a(30) = 3 since number of zeros in 302928272625242322212019181716151413121110987654321 is 3. (This example implies offset = 1.)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023.

Cf. A055640, A055641, A102669-A102685, A117804.

a061217 n = a061217_list !! (n-1)
a061217_list = scanl1 (+) $ map a055641 [1..]
-- Reinhard Zumkeller, Oct 27 2013

Table[Count[Flatten[IntegerDigits/@Table[x-n,{n,0,x-1}]],0],{x,110}] (* Harvey P. Dale, Aug 10 2011 *)

a(n) = my(m=logint(n,10)); (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2) * sum(j=1, m+1, (n\10^j * (2*n+2 - (1 + n\10^j) * 10 ^ j) - floor(n/10^j+9/10) * (2*n+2 + ((4/5 - floor(n / 10^j + 9 / 10))*10^j)))) \\ adapted from formula by Hieronymus Fischer \\ David A. Corneth, Jan 23 2019

From Hieronymus Fischer, Jun 12 2012: (Start)
a(n) = (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2)*Sum_{j=1..m+1} (floor(n/10^j)*(2*n + 2 - (1 + floor(n/10^j))*10^j) - floor(n/10^j + 9/10)*(2*n + 2 + (4/5 - floor(n/10^j + 9/10))*10^j)), where m=floor(log_10(n)).
a(n) = A117804(n+1) - (n+1)*A054640(n) + (1/2)*Sum_{j=1..m+1} ((floor(n/10^j + 9/10)^2 - floor(n/10^j)^2)*10^j - (4/5*floor(n/10^j + 9/10) + floor(n/10^j))*10^j), where m=floor(log_10(n)).
a(10^m-1) = m*10^(m-1) - (10^m-1)/9.
(This is the total number of zeros occurring in all the numbers 1..10^m-1 or numbers with <= m places excluding zero.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (1-x^10^j)*x^10^(j+1)/(1-x^10^(j+1)). (End)

Corrected and extended by Patrick De Geest, Jun 05 2001

Offset changed to 1 by Hieronymus Fischer, Jun 12 2012

A102679 Number of digits >= 7 in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007093 (numbers in base 7). - Bernard Schott, Feb 12 2023

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

Cf. A007093, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102680.

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(p(n),n=0..125); # Emeric Deutsch, Feb 23 2005

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

A102681 Number of digits >= 8 in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007094 (numbers in base 8). - Bernard Schott, Feb 18 2023

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

Cf. A007094, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102682.

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(p(n),n=0..120); # Emeric Deutsch, Feb 23 2005

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

A249304 Number of zeros in row n of triangle A249133.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 5, 3, 7, 13, 13, 11, 13, 15, 13, 7, 15, 29, 29, 27, 29, 31, 29, 23, 29, 39, 37, 31, 33, 35, 29, 15, 31, 61, 61, 59, 61, 63, 61, 55, 61, 71, 69, 63, 65, 67, 61, 47, 61, 87, 85, 79, 81, 83, 77, 63, 73, 91, 85, 71, 73, 75, 61, 31, 63, 125, 125

Offset: 0

Reinhard Zumkeller, Nov 14 2014

nonn

a(n) = (number of even terms in row n of triangle A249095) = A023416(A249184(n)) = A055641(A249183(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A249133, A048967, A249095, A023416, A055641, A249183, A249184.

a249304 n = if n == 0 then 0 else a048967 n + a048967 (n - 1)

Join[{0}, Plus @@@ Partition[Table[n + 1 - 2^DigitCount[n, 2, 1], {n, 0, 100}], 2, 1]] (* Amiram Eldar, Jul 28 2023 *)

a(n) = A048967(n) + A048967(n-1) for n > 0.

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

Original entry on oeis.org

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

A249183 a(n) = row n of triangle A249133, concatenated.

Original entry on oeis.org

1, 111, 11011, 1110111, 110101011, 11100000111, 1101100011011, 111011101110111, 11010101010101011, 1110000000000000111, 110110000000000011011, 11101110000000001110111, 1101010110000000110101011, 111000001110000011100000111, 11011000110110001101100011011

Offset: 0

Reinhard Zumkeller, Nov 14 2014

nonn

			.   0:                                  1
.   1:                                 111
.   2:                                11011
.   3:                               1110111
.   4:                              110101011
.   5:                             11100000111
.   6:                            1101100011011
.   7:                           111011101110111
.   8:                          11010101010101011
.   9:                         1110000000000000111
.  10:                        110110000000000011011
.  11:                       11101110000000001110111
.  12:                      1101010110000000110101011
.  13:                     111000001110000011100000111
.  14:                    11011000110110001101100011011
.  15:                   1110111011101110111011101110111
.  16:                  110101010101010101010101010101011
.  17:                 11100000000000000000000000000000111
.  18:                1101100000000000000000000000000011011
.  19:               111011100000000000000000000000001110111
.  20:              11010101100000000000000000000000110101011
.  21:             1110000011100000000000000000000011100000111
.  22:            110110001101100000000000000000001101100011011
.  23:           11101110111011100000000000000000111011101110111
.  24:          1101010101010101100000000000000011010101010101011
.  25:         111000000000000011100000000000001110000000000000111
.  26:        11011000000000001101100000000000110110000000000011011
.  27:       1110111000000000111011100000000011101110000000001110111
.  28:      110101011000000011010101100000001101010110000000110101011
.  29:     11100000111000001110000011100000111000001110000011100000111
.  30:    1101100011011000110110001101100011011000110110001101100011011
.  31:   111011101110111011101110111011101110111011101110111011101110111
.  32:  11010101010101010101010101010101010101010101010101010101010101011 .

Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Cf. A249133, A249184 (decimal), A007088, A006943.

a249183 = foldr (\b v -> 10 * v + b) 0 . a249133_row

a[n_] := FromDigits[Mod[Riffle[Binomial[n, Range[0, n]], Binomial[n - 1, Range[0, n - 1]]], 2]]; Array[a, 15, 0] (* Amiram Eldar, Jul 28 2023 *)

a(n) = Sum_{k=0..2*n} A249133(n,k)*10^k.
a(n) = A007088(A249184(n));
A055641(a(n)) = A249304(n).

A075877 Powering the decimal digits of n (left-associative).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 1

Reinhard Zumkeller, Oct 16 2002

See A256229 for the (maybe more natural) "right-associative" variant, a(xyz)=x^(y^z). a(n) = A256229(n) for n < 212 (up to 210, according to the 2nd formula which also holds for A256229), but (2^1)^2 = 4 while 2^(1^2) = 1. - M. F. Hasler, Mar 22 2015

			a(253) = (2^5)^3 = 32^3 = 32768.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A007954.

Cf. A185969, A248907, A256179.

a075877 n = if n < 10 then n else a075877 (n `div` 10) ^ (n `mod` 10)
-- Reinhard Zumkeller, May 27 2013

A075877(n)=if(n>9,A075877(n\10)^(n%10),n) \\ M. F. Hasler, Mar 19 2015

a(n) = if n < 10 then n else a(floor(n\10))^(n mod 10).
a(n) = 1 iff the initial digit is 1 or n contains a 0 (i.e., A055641(n) > 0 or A000030(n) = 1);
a(A011540(n)) = 1.
a(n) = A133500(n) for n <= 99. - Reinhard Zumkeller, May 27 2013

Formula corrected by Reinhard Zumkeller, May 27 2013

Edited by M. F. Hasler, Mar 22 2015

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

A356859 a(n) is the number of zero digits in the product of the first n numbers not divisible by 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 2, 4, 1, 2, 2, 2, 6, 5, 2, 3, 5, 4, 2, 5, 3, 4, 6, 4, 3, 8, 3, 3, 4, 8, 9, 6, 3, 5, 9, 6, 10, 9, 7, 4, 11, 10, 10, 8, 13, 9, 5, 8, 8, 11, 7, 8, 10, 13, 11, 10, 12, 11, 13, 13, 16, 6, 16, 10, 21, 17

Offset: 0

Stefano Spezia, Sep 01 2022

Cf. A047201, A055641, A356858.

Cf. A356860 (number of digits), A356861 (number of nonzero digits).

Table[Count[IntegerDigits[Product[Floor[(5i-1)/4], {i,n}]], 0], {n,0,80}]

from math import prod
def a(n): return str(prod((5*k-1)//4 for k in range(1, n+1))).count("0")
print([a(n) for n in range(81)]) # Michael S. Branicky, Sep 01 2022

a(n) = A055641(A356858(n)).

cp's OEIS Frontend

A316867 Number of times 6 appears in decimal expansion of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A061217 Number of zeros in the concatenation n(n-1)(n-2)(n-3)...321.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A102679 Number of digits >= 7 in decimal representation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A102681 Number of digits >= 8 in decimal representation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A249304 Number of zeros in row n of triangle A249133.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A249183 a(n) = row n of triangle A249133, concatenated.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula