A202267 Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0.

0, 1, 2, 3, 5, 7, 10, 11, 12, 13, 15, 17, 20, 21, 22, 23, 25, 27, 30, 31, 32, 33, 35, 37, 50, 51, 52, 53, 55, 57, 70, 71, 72, 73, 75, 77, 100, 101, 102, 103, 105, 107, 110, 111, 112, 113, 115, 117, 120, 121, 122, 123, 125, 127, 130, 131, 132, 133, 135, 137, 150

Offset

1,3

Comments

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)=0,1,2,3,5,7 for k=0..5. - Hieronymus Fischer, May 30 2012

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.

Formula

From Hieronymus Fischer, May 30 2012: (Start)

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

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

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

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

a(4*6^n+1) = 5*10^n.

a(5*6^n+1) = 7*10^n.

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

a(n) < 10^log_6(n-1) else.

a(n) = A007092(n-1) iff the digits of A007092(n-1) are <= 3, a(n)>A007092(n-1), else.

a(n) <= A084984(n), equality holds if the representation of n-1 as a base-6 number only has digits 0 or 1.

G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j *(1-x^6^j)* (1 + 2x^6^j + 3(x^2)^6^j + 5(x^3)^6^j + 7(x^4)^6^j)/(1-x^6^(j+1)).

Also: g(x) = (x/(1-x))*(h_(6,1)(x) + h_(6,2)(x) + h_(6,3)(x) + 2*h_(6,4)(x) + 2*h_(6,5)(x) - 7*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) = 4.945325883472729555972742252181522711968119529132581193614012706741310832798... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Example

a(1000) = 5353.
a(10^4) = 115153
a(10^5) = 2070753.
a(10^6) = 33233353.

Mathematica

Union[Flatten[FromDigits/@Tuples[{0, 1, 2, 3, 5, 7}, 3]]] (* Harvey P. Dale, Mar 11 2015 *)

Crossrefs

Supersequence of A001742 and A046034.

Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202268 (numbers in which all digits are nonprimes excluding 0), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).

Cf. A007092, A001743, A001744, A193238.

Sequence in context: A323013 A163975 A267521 * A125975 A046758 A121232

Adjacent sequences: A202264 A202265 A202266 * A202268 A202269 A202270

Keyword

nonn,base,easy

Author

Jaroslav Krizek, Dec 25 2011

Extensions

Examples added by Hieronymus Fischer, May 30 2012

Status

approved