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

Author

Jaroslav Krizek, Dec 25 2011

Keywords

• nonn
• base
• easy

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

Examples

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

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

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.

Programs

Mathematica

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

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

Extensions

Examples added by Hieronymus Fischer, May 30 2012