A029581 Numbers in which all digits are composite.

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 89, 94, 96, 98, 99, 444, 446, 448, 449, 464, 466, 468, 469, 484, 486, 488, 489, 494, 496, 498, 499, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 689, 694, 696, 698, 699, 844, 846, 848

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

If n is represented as a zerofree base-4 number (see A084544) according to n=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)=4,6,8,9 for k=1..4. - Hieronymus Fischer, May 30 2012

Examples

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 88649.
a(10^4) = 6468989
a(10^5) = 449466489. (End)
		

Links

• Hieronymus Fischer, 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. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

Programs

Magma

[n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

Mathematica

Table[FromDigits/@Tuples[{4, 6, 8, 9}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

Formula

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

a(n) = Sum_{j=0..m-1} (2*b(j) mod 8 + 4 + floor(b(j)/4) - floor((b(j)+1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).

Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.

Special values:

a(1*(4^n-1)/3) = 4*(10^n-1)/9.

a(2*(4^n-1)/3) = 2*(10^n-1)/3.

a(3*(4^n-1)/3) = 8*(10^n-1)/9.

a(4*(4^n-1)/3) = 10^n-1.

a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.

a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.

a(n) > 2*A084544(n).

Lower and upper limits:

lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.

lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.

where 10^log_4(n) = n^1.66096404744...

G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(4 + 6z(j) + 8*z(j)^2 + 9*z(j)^3)/(1-z(j)^4), where z(j) = x^4^j.

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

Sum_{n>=1} 1/a(n) = 1.039691381254753739202528087006945643166147087095114911673083135126969046250... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Extensions

Offset corrected by Arkadiusz Wesolowski, Oct 03 2011