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

Author

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

Keywords

• base
• easy
• nonn

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

Examples

			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.
		

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.

Crossrefs

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

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

Programs

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

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 *)

PARI

is(n)=isprime(eval(Vec(Str(n))))==0 \\ Charles R Greathouse IV, Feb 20 2012

PARI

my(table=[0,1,4,6,8,9]); \
a(n) = fromdigits([table[d+1] |d<-digits(n-1,6)]); \\ Kevin Ryde, May 27 2025

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+10.

a(n) > 10^log_6(n-1) for 2*6^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

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