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

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..]
-- Reinhard Zumkeller, Jul 19 2011
(Magma) [n: n in [0..169] | forall{d: d in [2, 3, 5, 7] | d notin Set(Intseq(n))}];  // Bruno Berselli, Jul 19 2011
(PARI) is(n)=isprime(eval(Vec(Str(n))))==0 \\ Charles R Greathouse IV, Feb 20 2012

Crossrefs

Cf. A046034, A034844 (primes), A118950, A193238.

Cf. A007092, A029581, A017042, A001743, A001744, A014263, A202267, A202268.

Sequence in context: A047748 A299540 A085558 * A104499 A338941 A137353

Adjacent sequences: A084981 A084982 A084983 * A084985 A084986 A084987

Keyword

base,nonn

Author

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 27 2003

Extensions

0 added by N. J. A. Sloane, Feb 02 2009

100 added by Arkadiusz Wesolowski, Mar 10 2011

Examples for n>=10^3 added by Hieronymus Fischer, May 30 2012

Status

approved