A162699 -id:A162699 - 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

A110551 Period 6: repeat [1, 3, 5, 5, 3, 1].

Original entry on oeis.org

1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1

Offset: 0

Paul Barry, Jul 26 2005

a(n) = A162699(n+1) (Modd 7) = A204453(A162699(n+1)), n>=0, where the nonnegative members of the seven residue classes Mod 7 (not to be confused with mod 7), called [m] for m=0..6, are given in the array A113807, if there the last row, starting with 7 is taken as class [0] after adding a 0 in front. Here only the classes [1], [3] and [5] are relevant. For Modd n residue classes see a comment on A203571. [Wolfdieter Lang, Feb 09 2012]

Continued fractions expansion of (8+sqrt(905))/29 = 1.3132144107925.. - R. J. Mathar, Mar 08 2012

			Modd 7 classes for positive odd numbers reduced mod 7: a(3)=5 because A162699(4)=9 (the fourth positive odd number not divisible by 7), and 9 is a member of the Modd 7 class [5] = {5,9,19,23,...}.
A162699: 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27,...
Modd 7:  1, 3, 5, 5,  3,  1,  1,  3,  5,  5,  3,  1,... [_Wolfdieter Lang_, Feb 09 2012]

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. A001045, A105198, A110549, A110550, A113807, A162699, A203571, A204453.

&cat [[1, 3, 5, 5, 3, 1]^^30]; // Wesley Ivan Hurt, Jun 29 2016

A110551:=n->[1, 3, 5, 5, 3, 1][(n mod 6)+1]: seq(A110551(n), n=0..100); # Wesley Ivan Hurt, Jun 29 2016

PadRight[{}, 100, {1, 3, 5, 5, 3, 1}] (* Wesley Ivan Hurt, Jun 29 2016 *)

x='x+O('x^50); Vec((1+x+x^2)/((1-x)*(x^2-x+1))) \\ G. C. Greubel, Aug 31 2017

From R. J. Mathar, Oct 15 2014: (Start)
G.f.: ( 1+x+x^2 ) / ( (1-x)*(x^2-x+1) ).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = 3 + 2*sin(Pi*n/3)/sqrt(3) - 2*cos(Pi*n/3).
a(n) = A001045(n+2) mod 6. (End)
From Wesley Ivan Hurt, Jun 29 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) for n>2. (End)

A160545 Numbers coprime to 21.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 50, 52, 53, 55, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 100, 101, 103, 104, 106, 107, 109, 110, 113, 115, 116

Offset: 1

Zerinvary Lajos, May 18 2009

The asymptotic density of this sequence is 4/7. - Amiram Eldar, Oct 23 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A162699, A045572.

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1}, {1,2,4,5,8,10,11,13,16,17,19,20,22},67] (* Ray Chandler, Jul 15 2015 *)
Select[Range[100], CoprimeQ[21, #] &] (* Amiram Eldar, Oct 23 2020 *)

[i for i in range(0,128) if gcd(21, i) == 1]

a(n+12) = a(n) + 21.
a(n) = 21*floor((n-1)/12)+f(n)+floor(f(n)/2)+2^floor(f(n)/4)+floor(((n+5) mod 12)/11)+floor(((n+3) mod 12)/11), where f(n)= (n-1) mod 12. - Gary Detlefs, Sep 22 2013
G.f.: x*(1+x+2*x^2+x^3+3*x^4+2*x^5+x^6+2*x^7+3*x^8+x^9+2*x^10+x^11+x^12) / ( (1+x) *(1+x^2) *(1+x+x^2) *(x^2-x+1) *(x^4-x^2+1) *(x-1)^2 ). - R. J. Mathar, Sep 27 2014

Definition corrected by Leroy Quet, Jun 19 2009

A204454 Odd numbers not divisible by 11.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131

Offset: 1

Wolfdieter Lang, Jan 24 2012

Up to a(45) this sequence coincides with A029740, but 101 is not in A029740.
This sequence is the fourth member of the family of sequences of odd numbers not divisible by a given odd prime p. For p = 3, 5, and 7 these sequences are A007310, A045572, and A162699, respectively.  The formula is
  a(p;n) = 2*n+1 + 2*floor((n-(p+1)/2)/(p-1)), n>=1, p an odd prime. If one puts a(p;0):=0, the o.g.f. is
  G(p;x) = (x/((1-x^(p-1))*(1-x)))*(1 + 2*sum(x^k,k=1..(p-3)/2) + 4*x^((p-1)/2) + 2*sum(x^((p-1)/2+k),k=1..(p-3)/2) + x^(p-1)).
See the array A204456 with the coefficients of the numerator polynomials of these o.g.f.s.
This sequence gives also the numbers relatively prime to 2 and 11.
Another formula is a(p;n) = 2*n-1 + 2*floor(( n-(p-3)/2)/(p-1)), n>=1. From the rows of the array A204456 for the o.g.f. one can show first: a(p;n) = n + sum(floor((n+p-3-k)/(p-1)),k=1..(p-3)/2) + 3*floor((n+(p-3)/2)/(p-1)) + sum(floor((n+(p-3)/2-k)/(p-1)),k=1..(p-1)/2), p an odd prime, n>=1. - Wolfdieter Lang, Jan 26 2012
Recurrences for odd p: a(p;n) = a(p;n-(p-1)) + 2*p. For first differences: a(p;n) = a(p;n-1) + a(p;n-p+1) - a(p;n-p), n>=p, and inputs a(p;0):=-1 (here not 0) and a(p;k) for k=1,...,p-1. See the formula sections of the A-numbers for the instances p = 3, 5, and 7 for the contributions from Zak Seidov and R. J. Mathar. From this recurrence follows the o.g.f. (starting with x^1) directly. Above it has been found from the formula for a(p;n). Here the coefficients of the numerator polynomial of the o.g.f. (besides the 1s for x^1 and x^p) arise as first differences of the input members of the {a(p;n)} sequence. - Wolfdieter Lang, Jan 27 2012
Numbers coprime to 22. The asymptotic density of this sequence is 5/11. - Amiram Eldar, Oct 20 2020

			2*floor((n-6)/10), n>=0, is the sequence (the exponent of a number indicates how many times this number appears consecutively): (-2)^6 0^10 2^10 4^10 ... By adding these numbers to 2*n+1, n>=0, one obtains -1 for n=0 and a(n) for n>=1. The o.g.f is computed from this sum, but adjusted such that one obtains a vanishing a(0).
Recurrences: 31 = a(15) = a(5) + 2*11 = 9 + 22. a(15) = a(14) + a(5) - a(4) = 29 + 9 - 7 = 31. - _Wolfdieter Lang_, Jan 27 2012

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

Cf. A007310, A045572, A162699, A204456, A204457, A204458.

Complement[Range[1, 131, 2], 11Range[11]] (* Alonso del Arte, Jan 24 2012 *)

a(n)=2*n+(n-6)\10*2+1 \\ Charles R Greathouse IV, Jan 24 2012

a(n) = 2*n+1 + 2*floor((n-6)/10), n>=1. Note that this is -1 for n=0, but the following o.g.f. uses a(0)=0.
O.g.f: x*(1+2*x+2*x^2+2*x^3+2*x^4+4*x^5+2*x^6+2*x^7+2*x^8+2*x^9+x^10)/((1-x^10)*(1-x)). See the comment above for p=11.
a(n) = n + sum(floor((n+9-k)/10),k=1..4) + 3*floor((n+4)/10) + sum(floor((n+4-k)/10),k=1..5) = n + (n-1) + 2*floor((n+4)/10), n>=1. See the line m=5, p=11 of the array A204456, and the general formula given in a comment above. - Wolfdieter Lang, Jan 26 2012
Recurrences: a(n) = a(n-10) + 2*11. First differences: a(n) = a(n-1) + a(n-10) - a(n-11), n>=11, and inputs a(p;0):=-1 ( here not 0) and a(p;k) for k=1,...,10. See the general comment above. - Wolfdieter Lang, Jan 27 2012

A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 22, 24, 26, 30, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 100, 102, 104, 105, 106, 108, 110, 114, 116, 118, 119, 120, 122, 124, 126, 130, 132

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003591 and A162699.
Analogous to A081062 and A105115 that apply to A120944(1) and A120944(2), respectively.
This sequence applies to A120944(3).

			6 is in the sequence since gcd(6, 14) = 2 and 6 does not divide 14^e with integer e >= 0.
2 is not in the sequence since 2 | 14.
4 is not in the sequence since 4 | 14^2.
3 and 5 are not in the sequence since they are coprime to 14.

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

Cf. A003591, A081062, A105115, A120944, A162699.

With[{nn = 132, k = 14}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A204456 Coefficient array of numerator polynomials of the o.g.f.s for the sequence of odd numbers not divisible by a given prime.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 2, 4, 2, 1, 1, 2, 2, 4, 2, 2, 1, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Wolfdieter Lang, Jan 24 2012

The row length sequence of this array is p(m) = A000040(m) (the primes).
Row m, for m >= 1, lists the coefficients of the numerator polynomials N(p(m);x) = Sum_{k=0..p(m)-1} a(m,k)*x^k for the o.g.f. G(p(m);x) = x*N(p(m);x)/((1-x^(p(m)-1))*(1-x)) for the sequence a(p(m);n) of odd numbers not divisible by p(n). For m=1 one has a(2;n)=2*n-1, n >= 1, and for m > 1 one has a(p(m);n) = 2*n+1 + floor((n-(p(m)+1)/2)/(p(m)-1)), n >= 1, and a(p(m);0):=0. See A204454 for the m=5 sequence a(11;n), also for more details.
The rows of this array are symmetric. For m > 1 they are symmetric around the central 4.
The first (p(m)+1)/2 numbers of row number m, for m >= 2, are given by the first differences of the corresponding sequence {a(p(m);n)}, with a(p(m),0):=0. See a formula below. The proof is trivial for m=1, and clear for m >= 2 from a(p(m);n), for n=0,...,(p(m)+1)/2, which is {0,1,3,...,p-2,p+2}. - Wolfdieter Lang, Jan 26 2012

			The array starts
m,p\k  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
1,2:   1  1
2,3:   1  4  1
3,5:   1  2  4  2  1
4,7:   1  2  2  4  2  2  1
5,11:  1  2  2  2  2  4  2  2  2  2  1
6,13:  1  2  2  2  2  2  4  2  2  2  2  2  1
7,17:  1  2  2  2  2  2  2  2  4  2  2  2  2  2  2  2  1
...
N(p(4);x) = N(7;x) = 1 + 2*x + 2*x^2 + 4*x^3 + 2*x^4 + 2*x^5 + x^6 = (1+x^2)*(1+2*x+x^2+2*x^3+x^4).
G(p(4);x) = G(7;x) = x*N(7;x)/((1-x^6)*(1-x)), the o.g.f. of
A162699. Compare this with the o.g.f. given there by _R. J. Mathar_, where the numerator is factorized also.
First difference rule: m=4: {a(7;n)} starts {0,1,3,5,9,...},
the first differences are {1,2,2,4,...}, giving the first (7+1)/2=4 entries of row number m=4 of the array. The other entries follow by symmetry. - _Wolfdieter Lang_, Jan 26 2012

Cf. A000040, A005408 (p=2), A007310 (p=3), A045572 (p=5), A162699 (p=7), A204454 (p=11).

a(m,k) = [x^k]N(p(m);x), m>=1, k=0,...,p(m)-1, with the numerator polynomial N(p(m);x) for the o.g.f. G(p(m);x) of the sequence of odd numbers not divisible by the m-th prime p(m)=A000040(m). See the comment above.
Row m has the number pattern (exponents on a number indicate how many times this number appears consecutively):
  m=1, p(1)=2: 1 1, and for m>=2:
  m, p(m): 1 2^((p(m)-3)/2) 4 2^((p(m)-3)/2) 1.
a(m,k) = a(p(m);k+1) - a(p(m);k), m>=2, k=0,...,(p(m)-1)/2,
with the corresponding sequence {a(p(m);n)} of the odd numbers not divisible by p(m), with a(p(m);0):=0. For m=1: a(1,0) = a(2;1)-a(2;0). By symmetry around the center: a(m,(p(m)-1)/2+k) = a(m,(p(m)-1)/2-k), k=1,...,(p(m)-1)/2, m>=2. For m=1: a(1,1)=a(1,0). See a comment above. - Wolfdieter Lang, Jan 26 2012

cp's OEIS Frontend

A229829 Numbers coprime to 15.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

A110551 Period 6: repeat [1, 3, 5, 5, 3, 1].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A160545 Numbers coprime to 21.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A204454 Odd numbers not divisible by 11.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A316991 Numbers m such that 1 < gcd(m, 14) < m and m does not divide 14^e for e >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A204456 Coefficient array of numerator polynomials of the o.g.f.s for the sequence of odd numbers not divisible by a given prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula