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A029581 All digits are composite. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences.

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)

EXAMPLE

From Hieronymus Fischer, May 30 2012: (Start)

a(1000) = 88649.

a(10^4) = 6468989

a(10^5) = 449466489. (End)

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

Sequence in context: A001744 A113624 A113591 * A202262 A202266 A232541

Adjacent sequences: A029578 A029579 A029580 * A029582 A029583 A029584

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane

EXTENSIONS

Offset corrected by Arkadiusz Wesolowski, Oct 03 2011

STATUS

approved

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Last modified November 26 14:12 EST 2022. Contains 358362 sequences. (Running on oeis4.)