A235933 -id:A235933 - cp's OEIS frontend

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

Gary Detlefs, Oct 01 2013

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

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).

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.

[n: n in [1..120] | IsOne(GCD(n,15))]; // Bruno Berselli, Oct 01 2013

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

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 *)

[i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013

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

A236206 Numbers not divisible by 3, 5 or 7.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 17, 19, 22, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 118, 121, 122, 124, 127, 128, 131, 134, 136

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers whose odd part is 11-rough: products of terms of A008364 and powers of 2 (terms of A000079). - Peter Munn, Aug 03 2020

Numbers coprime to 105. The asymptotic density of this sequence is 16/35. - Amiram Eldar, Oct 23 2020

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

Subsequences: A000079, A008364.
Intersection of any 2 of A160545, A229829, A235933.
Other sequences with similar definitions: A007775, A236217.

Select[Range[300], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 7] > 0 &] (* T. D. Noe, Feb 05 2014 *)
Select[Range[300],Or@@Divisible[#,{3,5,7}]==False&] (* Harvey P. Dale, Mar 13 2014 *)
Select[Range[150], CoprimeQ[105, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-48) - a(n-49). - Amiram Eldar, Oct 23 2020

A307589 Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35^e for e >= 0.

Original entry on oeis.org

10, 14, 15, 20, 21, 28, 30, 40, 42, 45, 50, 55, 56, 60, 63, 65, 70, 75, 77, 80, 84, 85, 90, 91, 95, 98, 100, 105, 110, 112, 115, 119, 120, 126, 130, 133, 135, 140, 145, 147, 150, 154, 155, 160, 161, 165, 168, 170, 180, 182, 185, 189, 190, 195, 196, 200, 203, 205

Offset: 1

Michael De Vlieger, Aug 22 2019

Complement of the union of A003595 and A235933.

Analogous to A081062 and A105115 for terms 1 and 2 of A120944. This sequence applies to A120944(6) = 35.

			10 is in the sequence since gcd(10, 35) = 5 and 10 does not divide 35^e with integer e >= 0.
2 is not in the sequence since 2 is coprime to 35.
7 is not in the sequence since 7 | 35.
25 is not in the sequence since 25 | 35^2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A003595, A081062, A105115, A120944, A235933, A306999.

With[{nn = 205, k = 35}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

is(n)=gcd(n,35)>1 && n/5^valuation(n,5)/7^valuation(n,7)>1 \\ Charles R Greathouse IV, Sep 07 2022

a(n) = 35n/11 + O(log^2 n). - Charles R Greathouse IV, Sep 07 2022

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A229829 Numbers coprime to 15.

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A236206 Numbers not divisible by 3, 5 or 7.

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A307589 Numbers m such that 1 < gcd(m, 35) < m and m does not divide 35^e for e >= 0.

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