A160545 -id:A160545 - 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

A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 28, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, 66, 69, 70, 72, 75, 77, 78, 84, 87, 90, 91, 93, 96, 98, 99, 102, 105, 108, 111, 112, 114, 117, 119, 120, 123, 126, 129, 132, 133, 135, 138, 140, 141, 144, 150, 153, 154, 156, 159, 161

Offset: 1

Michael De Vlieger, Aug 22 2019

Complement of the union of A003594 and A160545.

Analogous to A081062 and A105115 regarding terms 1 and 2 of A120944, respectively. This sequence applies to A120944(5) = 21.

			6 is in the sequence since gcd(6, 21) = 3 and 6 does not divide 21^e with integer e >= 0.
5 is not in the sequence since it is coprime to 21.
3 is not in the sequence since 3 | 21.
9 is not in the sequence since 9 | 21^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003594, A081062, A105115, A120944, A160545.

filter:= proc(n) local g;
  g:= igcd(n,21);
  if g = 1 or g = n then return false fi;
  3^padic:-ordp(n,3)*7^padic:-ordp(n,7) < n
end proc:
select(filter, [$1..200]); # Robert Israel, Aug 28 2019

With[{nn = 161, k = 21}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A229973 Numbers coprime to 39.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 56, 58, 59, 61, 62, 64, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 100, 101, 103, 106

Offset: 1

Gary Detlefs, Oct 04 2013

Numbers not divisible by 3 or 13.
For n from 1 to 24, a(n) mod 39-n - floor(11*n/25)-2*floor(n/8) has a period of 24, consisting of all zeros except a -2 at indices 8, 16, and 24.
The asymptotic density of this sequence is 8/13. - 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 (2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1).

Cf. A160545, A229829, A229968.

for n from 1 to 50 do if n mod 3<>0 and n mod 13<>0 then print(n) fi od

CoefficientList[Series[(x^22 + x^20 + x^18 + x^16 + 2 x^14 - x^12 + 3 x^11 - x^10 + 2 x^8 + x^6 + x^4 + x^2 + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + 1) (x^4 - x^2 + 1) (x^4 + 1) (x^8 - x^4 + 1)), {x, 0, 80}], x] (* Vincenzo Librandi, Oct 08 2013 *)
Select[Range[100], CoprimeQ[39, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n+24) = a(n) + 39.
a(n) = 39*floor((n-1)/24) + f(n) + floor(11*f(n)/25) + 2*floor(f(n)/8) - 2*floor(((n-1)mod 8)/7) + 40*floor(f(n-1)/23), where f(n) = n mod 24.
G.f.: x*(x^22+x^20+x^18+x^16+2*x^14-x^12+3*x^11-x^10+2*x^8+x^6+x^4+x^2+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)*(x^4+1)*(x^8-x^4+1)). - Colin Barker, Oct 07 2013

More terms from Colin Barker, Oct 07 2013

a(34) corrected by Vincenzo Librandi, Oct 08 2013

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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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A306999 Numbers m such that 1 < gcd(m, 21) < m and m does not divide 21^e for e >= 0.

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A229973 Numbers coprime to 39.

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