A229829 Numbers coprime to 15.

1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119

Offset

1,2

Comments

A001651 INTERSECT A047201.

a(n) - 15*floor((n-1)/8) - 2*((n-1) mod 8) has period 8, repeating [1,0,0,1,0,1,1,0].

Numbers whose odd part is 7-rough: products of terms of A007775 and powers of 2 (terms of A000079). - Peter Munn, Aug 04 2020

The asymptotic density of this sequence is 8/15. - Amiram Eldar, Oct 18 2020

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Formula

a(n+8) = a(n) + 15.

a(n) = 15*floor((n-1)/8) +2*f(n) +floor(2*phi*(f(n+1)+2)) -2*floor(phi*(f(n+1)+2)), where f(n) = (n-1) mod 8 and phi=(1+sqrt(5))/2.

a(n) = 15*floor((n-1)/8) +2*f(n) +floor((2*f(n)+5)/5) -floor((f(n)+2)/3), where f(n) = (n-1) mod 8.

From Bruno Berselli, Oct 01 2013: (Start)

G.f.: x*(1 +x +2*x^2 +3*x^3 +x^4 +3*x^5 +2*x^6 +x^7 +x^8) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). -

a(n) = a(n-1) +a(n-8) -a(n-9) for n>9. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(7 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 2021

Maple

for n from 1 to 500 do if n mod 3<>0 and n mod 5<>0 then print(n) fi od

Mathematica

Select[Range[120], GCD[#, 15] == 1 &] (* or *) t = 70; CoefficientList[Series[(1 + x + 2 x^2 + 3 x^3 + x^4 + 3 x^5 + 2 x^6 + x^7 + x^8)/((1 - x)^2 (1 + x) (1 + x^2) (1 + x^4)) , {x, 0, t}], x] (* Bruno Berselli, Oct 01 2013 *)
Select[Range[120], CoprimeQ[#, 15]&] (* Harvey P. Dale, Oct 31 2013 *)

Prog

(Magma) [n: n in [1..120] | IsOne(GCD(n, 15))]; // Bruno Berselli, Oct 01 2013
(Sage) [i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013

Crossrefs

Lists of numbers coprime to other semiprimes: A007310 (6), A045572 (10), A162699 (14), A160545 (21), A235933 (35).

Cf. A008676, A078588, A113763, A160542, A160543, A160544.

Subsequence of: A001651, A047201.

Subsequences: A000079, A007775.

Sequence in context: A323108 A187418 A161989 * A000069 A344602 A333975

Adjacent sequences: A229826 A229827 A229828 * A229830 A229831 A229832

Keyword

nonn,easy

Author

Gary Detlefs, Oct 01 2013

Status

approved