A235583 Numbers not divisible by 2, 5 or 7.

1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 127, 129, 131, 137, 139, 141, 143, 149, 151, 153, 157, 159, 163, 167, 169, 171, 173, 177, 179, 181, 183

Offset: 1

Author

Oleg P. Kirillov, Jan 12 2014

Keywords

• nonn
• easy

Comments

All primes, except 2, 5 and 7, are in this sequence. Any product of terms is also a term in the sequence. For example, a(2)a(4) = 3 * 11 = 33 = a(12). - Alonso del Arte, Jan 12 2014

In other words, numbers equivalent 1,3,9,...,69 modulo 70. This means the first differences of the sequence are 24-periodic. - Ralf Stephan, Jan 14 2014

Numbers coprime to 70. The asymptotic density of this sequence is 12/35. - Amiram Eldar, Oct 23 2020

Examples

			51 = 3 * 17, and gcd(51, 70) = 1, so it is in the sequence.
53 is prime, so it is in the sequence.
55 = 5 * 11, and gcd(55, 70) = 5, so it is not in the sequence.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1).

Crossrefs

Cf. A007775, A008364 (subsequence).

Programs

Mathematica

Select[Range[210], GCD[#, 70] == 1 &] (* Alonso del Arte, Jan 12 2014 *)
Select[Range[300], Mod[#, 2]>0 &&Mod[#, 5]>0 &&Mod[#, 7]>0&] (* Vincenzo Librandi, Feb 08 2014 *)

Formula

G.f.: x*(x^22 +3*x^21 +8*x^20 +7*x^19 +x^18-2*x^17 -x^16 +5*x^15 +10*x^14 +7*x^13 -x^12 -6*x^11 -x^10 +7*x^9 +10*x^8 +5*x^7 -x^6 -2*x^5 +x^4 +7*x^3 +8*x^2 +3*x +1) / ((x+1) *(x^2+1) *(x^2+x+1) *(x^4-x^2+1) *(x^4+1) *(x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Jan 12 2014