A055640 -id:A055640 - cp's OEIS frontend

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

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

A189398 a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n.

Original entry on oeis.org

2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 16, 48, 144, 432, 1296, 3888, 11664, 34992, 104976, 314928, 32

Offset: 1

Reinhard Zumkeller, May 03 2011

Not the same as A061509: a(n) = A061509(n) for n <= 100; a(101)=2^1*3^0*5^1=10 <> A061509(101)=2^1*3^1=6;
a(A052382(n)) = A000079(A000030(a052382(n))) = A061509(A052382(n));
a(A002275(n)) = A002110(n): a(n-th rep-unit) = n-th primorial;
a(n*A011557(k)) = a(n): trailing zeros don't matter;
A001221(a(n)) = A055640(n): number of distinct prime factors of a(n) = number of nonzero digits of n;
A001222(a(n)) = A007953(n): number of all prime factors of a(n) = sum of digits of n;
a(81312000) = 2^8*3^1*5^3*7^1*11^2*13^0*17^0*19^0 = 81312000, the smallest fixed point, is called the Meertens number.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Richard S. Bird, Functional Pearl: Meertens number, Journal of Functional Programming 8 (1), Jan 1998, 83-88.
Wikipedia, Meertens number

Cf. A000040.

import Data.Char (digitToInt)
import Data.List (findIndices)
a189398 n = product $ zipWith (^) a000040_list (map digitToInt $ show n)
-- Two computations of the Meertens number: the first is brute force,
meertens = map succ $ findIndices (\x -> a189398 x == x) [1..]
-- ... and the second is more efficient, from Bird reference, page 87:
meertens' k = [n | (n,g) <- candidates (0,1), n == g] where
  candidates        = concat . map (search pps) . tail . labels ps
  ps : pps          = map (\p -> iterate (p *) 1) $ take k a000040_list
  search [] x       = [x]
  search (ps:pps) x = x : concat (map (search pps) (labels ps x))
  labels ps (n,g)   = zip (map (10*n +) [0..9]) (chop $ map (g *) ps)
  chop              = takeWhile (< 10^k)
-- Time and space required, GHC interpreted, Mac OS X, 2.66 GHz:
-- for >head meertens: (466.87 secs, 254780027728 bytes);
-- for >meertens' 8:   (  0.28 secs,     62027124 bytes).

a:= n-> `if`(n=0, 1, ithprime(length(n))^irem(n, 10, 'm') *a(m)):
seq(a(n), n=1..110);  # Alois P. Heinz, May 04 2011

a[n_] := (p = Prime[Range[Length[d = IntegerDigits[n]]]]; Times @@ (p^d)); Array[a, 50] (* Jean-François Alcover, Jan 09 2016 *)

a(n)=my(d=digits(n),p=primes(#d)); prod(i=1,#d,p[i]^d[i]) \\ Charles R Greathouse IV, Aug 19 2014

from sympy import prime
from operator import mul
from functools import reduce
def A189398(n):
    return reduce(mul, (prime(i)**int(d) for i,d in enumerate(str(n),start=1)))
# Chai Wah Wu, Aug 31 2014

# implementation using recursion
from sympy import prime
def _A189398(n):
    nlen = len(n)
    return _A189398(n[:-1])*prime(nlen)**int(n[-1]) if nlen > 1 else 2**int(n)
def A189398(n):
    return _A189398(str(n))
# Chai Wah Wu, Aug 31 2014

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

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

A356861 a(n) is the number of nonzero digits in the product of the first n numbers not divisible by 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 5, 6, 5, 8, 10, 10, 11, 12, 15, 14, 16, 19, 20, 20, 19, 23, 24, 25, 27, 24, 27, 31, 32, 32, 34, 38, 36, 40, 41, 40, 44, 47, 43, 50, 52, 53, 50, 51, 56, 61, 60, 58, 63, 61, 64, 67, 72, 67, 70, 72, 76, 72, 78, 84, 83, 85, 84, 90, 91, 91, 90

Offset: 0

Stefano Spezia, Sep 01 2022

Cf. A047201, A055640, A356858.

Cf. A356859 (number of zero digits), A356860 (number of digits).

Table[Length[Select[IntegerDigits[Product[Floor[(5i-1)/4], {i,n}]], Positive]], {n,0,68}]

from math import prod
def a(n): s = str(prod((5*k-1)//4 for k in range(1, n+1))); return len(s) - s.count("0")
print([a(n) for n in range(69)]) # Michael S. Branicky, Sep 01 2022

a(n) = A055640(A356858(n)).

A190311 Number of nonzero digits when writing n in base where place values are positive cubes, cf. A000433.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3

Offset: 0

Reinhard Zumkeller, May 08 2011

For n > 0: a(A000578(n)) = 1; for n > 1: a(A001093(n)) = 2;

a(n) <= A048766(n).

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

Cf. A190321, A055640.

a190311 n = g n $ reverse $ takeWhile (<= n) $ tail a000578_list where
  g _ []                 = 0
  g m (x:xs) | x > m     = g m xs
             | otherwise = signum m' + g r xs where (m',r) = divMod m x

A358993 a(n) is the number of nonzero digits in the product of the first n odd numbers not divisible by 5.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 6, 7, 9, 10, 12, 12, 13, 15, 18, 17, 20, 21, 22, 24, 25, 27, 29, 31, 32, 34, 34, 35, 40, 40, 45, 45, 45, 46, 47, 46, 52, 55, 55, 59, 58, 60, 59, 66, 66, 65, 69, 70, 74, 80, 79, 84, 75, 83, 90, 89, 87, 92, 95, 91, 95, 104, 98, 102, 110, 107

Offset: 0

Stefano Spezia, Dec 09 2022

Cf. A045572, A055640, A356861, A358990.

Cf. A358991 (number of zero digits), A358992 (number of digits).

Table[Length[Select[IntegerDigits[Product[2i+2Floor[(i-3)/4]+1, {i, n}]], Positive]], {n, 0, 66}]

a(n) = A055640(A358990(n)).

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

cp's OEIS Frontend

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

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A189398 a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n.

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

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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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A356861 a(n) is the number of nonzero digits in the product of the first n numbers not divisible by 5.

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A190311 Number of nonzero digits when writing n in base where place values are positive cubes, cf. A000433.

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A358993 a(n) is the number of nonzero digits in the product of the first n odd numbers not divisible by 5.

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

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