cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A204458 Odd numbers not divisible by 17.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141
Offset: 1

Views

Author

Wolfdieter Lang, Feb 07 2012

Keywords

Comments

For the general case of odd numbers not divisible by a prime see a comment on A204454. There the o.g.f.s and the formulas are given.
The numerator polynomial of the o.g.f. given below has coefficients 1,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,1. See the row no. 7 of the array A204456. The first nine numbers are the first differences of the sequence if one starts with a(0):=0. The remaining ones are obtained by mirroring around the central number 4.
Compare with A192861: certain numbers from here are missing there, like 35, 49, 53, 71, 89, 97, 99, .. and others are missing here like 51, 85, 119, ...
Numbers coprime to 34. The asymptotic density of this sequence is 8/17. - Amiram Eldar, Oct 20 2020

Links

Crossrefs

Cf. A204454 (also for more crossrefs), A204457.

Programs

  • Mathematica
    Select[Range[141], CoprimeQ[#, 34] &] (* Amiram Eldar, Oct 20 2020 *)

Formula

O.g.f.: x*(1 + x^16 + 2*x*(1+x^8)*(Sum_{k=0..6} x^k) + 4*x^8)/((1-x^16)*(1-x)). The denominator can be factored.
a(n) = 2*n-1 + 2*floor((n+7)/16) = 2*n+1 + 2*floor((n-9)/16), n>=1. Note that for n=0 this is -1, but for the o.g.f. with start x^0 one uses a(0)=0.
a(n) = a(n-1) + a(n-16) - a(n-17). - Wesley Ivan Hurt, Oct 20 2020

A204457 Odd numbers not divisible by 13.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 93, 95, 97, 99, 101, 103, 105, 107, 109
Offset: 1

Views

Author

Wolfdieter Lang, Feb 07 2012

Keywords

Comments

For the general case of odd numbers not divisible by primes see a comment on A204454, where the o.g.f.s and the formulas in terms of floor functions are given.
The numerator polynomial of the o.g.f. given in the formula section has coefficients 1,2,2,2,2,2,4,2,2,2,2,2,1, see row no. 6 of A204456. The first seven numbers are the first differences of the sequence, starting with a(0)=0. The other numbers are obtained by mirroring around the center.
Numbers coprime to 26. The asymptotic density of this sequence is 6/13. - Amiram Eldar, Oct 20 2020

Links

Crossrefs

Cf. A204454 and cross-references there; A204458.

Programs

  • Haskell
    a204457 n = a204457_list !! (n-1)
a204457_list = [x | x <- [1, 3 ..], mod x 13 > 0]
-- Reinhard Zumkeller, Feb 08 2012
  • Mathematica
    Select[Range[1,111,2],!Divisible[#,13]&] (* or *) With[{nn=111}, Complement[ Range[1,nn,2],13*Range[Floor[nn/13]]]] (* Harvey P. Dale, Jul 23 2013 *)
  • PARI
    a(n) = 2*n-1+(n+5)\12*2 \\ Charles R Greathouse IV, Feb 08 2012

Formula

O.g.f.: x*(1 + 2*(x+x^6)*(1+x+x^2+x^3+x^4) + 4*x^6 + x^12)/((1-x^12)*(1-x)). The denominator can be factored.
a(n) = 2*n-1 + 2*floor((n+5)/12) = 2*n+1 + 2*floor((n-7)/12), n>=1. Note that this is -1 for n=0, but the o.g.f. starting with x^0 has a(0)=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

Views

Author

Wolfdieter Lang, Jan 24 2012

Keywords

Comments

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

Examples

			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

Crossrefs

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

Formula

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

A206543 Period 10: repeat 1, 3, 5, 7, 9, 9, 7, 5, 3, 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1
Offset: 1

Views

Author

Wolfdieter Lang, Feb 09 2012

Keywords

Comments

For general Modd n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Modd 11 for the positive odd numbers not divisible by 11, which are given in A204454.
The underlying period length 22 sequence with offset 0 is P_11, also called Modd11, periodic([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]).

Examples

			Residue Modd 11 of the positive odd numbers not divisible by 11:
A204454: 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, ...
Modd 11: 1, 3, 5, 7, 9,  9,  7,  5,  3,  1,  1,  3,  5, ...

Links

Crossrefs

Cf. A000012 (Modd 3), A084101 (Modd 5), A110551 (Modd 7).

Programs

  • Mathematica
    PadRight[{},120,{1,3,5,7,9,9,7,5,3,1}] (* or *) LinearRecurrence[{2,-2,2,-2,1},{1,3,5,7,9},120] (* Harvey P. Dale, Oct 15 2017 *)
  • PARI
    a(n)=[1, 3, 5, 7, 9, 9, 7, 5, 3, 1][n%10+1] \\ Charles R Greathouse IV, Jul 17 2016

Formula

a(n) = A204454(n) (Modd 11) := Modd11(A204454(n)), with the periodic sequence Modd11 with period length 22 given in the comment section.
O.g.f.: x*(1+x^9+3*x*(1+x^7)+5*x^2*(1+x^5)+7*x^3*(1+x^3)+9*x^4*(1+x))/(1-x^10) = x*(1+x)*(1-x^5)/((1+x^5)*(1-x)^2).
Showing 1-4 of 4 results.

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