A084984 - OEIS

A084984

Numbers containing no prime digits.

0, 1, 4, 6, 8, 9, 10, 11, 14, 16, 18, 19, 40, 41, 44, 46, 48, 49, 60, 61, 64, 66, 68, 69, 80, 81, 84, 86, 88, 89, 90, 91, 94, 96, 98, 99, 100, 101, 104, 106, 108, 109, 110, 111, 114, 116, 118, 119, 140, 141, 144, 146, 148, 149, 160, 161, 164, 166, 168, 169

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Complement of A118950. - Reinhard Zumkeller, Jul 19 2011

If n-1 is represented as a base-6 number (see A007092) according to n-1=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n)= sum_{j=0..m} c(d(j))*10^j, where c(k)=0,1,4,6,8,9 for k=0..5. - Hieronymus Fischer, May 30 2012

LINKS

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

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

Index entries for 10-automatic sequences.

FORMULA

A193238(a(n)) = 0. - Reinhard Zumkeller, Jul 19 2011

a(n) >> n^1.285. - Charles R Greathouse IV, Feb 20 2012

From Hieronymus Fischer, May 30 and Jun 25 2012: (Start)

a(n) = ((2*b_m(n)+1) mod 10 + floor((b_m(n)+4)/5) - floor((b_m(n)+1)/5))*10^m + sum_{j=0..m-1} ((2*b_j(n))) mod 12 + floor(b_j(n)/6) - floor((b_j(n)+1)/6) + floor((b_j(n)+4)/6) - floor((b_j(n)+5)/6)))*10^j, where n>1, b_j(n)) = floor((n-1-6^m)/6^j), m = floor(log_6(n-1)).

Special values:

a(1*6^n+1) = 1*10^n.

a(2*6^n+1) = 4*10^n.

a(3*6^n+1) = 6*10^n.

a(4*6^n+1) = 8*10^n.

a(5*6^n+1) = 9*10^n.

a(2*6^n) = 2*10^n - 1.

a(n) = 10^log_6(n-1) for n=6^k+1, k>0.

Inequalities:

a(n) < 10^log_6(n-1) for 6^k+1<n<=2*6^k, k>0.

a(n) > 10^log_6(n-1) for 2*6^k<n<=6*6^k, k>=0.

a(n) <= 4*10^(log_6(n-1)-log_6(2)) = 1.641372618*10^(log_6(n-1)), equality holds for n=2*6^k+1, k>=0.

a(n) > 2*10^(log_6(n-1)-log_6(2)) = 0.820686309*10^(log_6(n-1)).

a(n) = A007092(n-1) iff the digits of A007092(n-1) are 0 or 1, a(n)>A007092(n-1), else.

a(n) >= A202267(n), equality holds if the representation of n-1 as a base-6 number has only digits 0 or 1.

Lower and upper limits:

lim inf a(n)/10^log_6(n) = 2/10^log_6(2) = 0.820686309, for n --> inf.

lim sup a(n)/10^log_6(n) = 4/10^log_6(2) = 1.641372618, for n --> inf.

where 10^log_6(n) = n^1.2850972089...

G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j * (1-x^6^j)*((1+x^6^j)^4 + 4(1+2x^6^j) * x^(3*6^j))/(1-x^6^(j+1)).

Also: g(x) = (x/(1-x))*(h_(6,1)(x) + 3*h_(6,2)(x) + 2*h_(6,3)(x) + 2*h_(6,4)(x) + h_(6,5)(x) - 9*h_(6,6)(x)), where h_(6,k)(x) = sum_{j>=0} 10^j*x^(k*6^j)/(1-x^6^(j+1)). (End)

Sum_{n>=2} 1/a(n) = 3.614028405471074989720026361356036456697082276983705341077940360653303099111... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

EXAMPLE

166 has digits 1 and 6 and they are nonprime digits.

a(1000) = 8686.

a(10^4) = 118186

a(10^5) = 4090986.

a(10^6) = 66466686.

MATHEMATICA

npdQ[n_]:=And@@Table[FreeQ[IntegerDigits[n], i], {i, {2, 3, 5, 7}}]; Select[ Range[ 0, 200], npdQ] (* Harvey P. Dale, Jul 22 2013 *)

PROG

(Haskell)

a084984 n = a084984_list !! (n-1)

a084984_list = filter (not . any (`elem` "2357") . show ) [0..]