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 1-5 of 5 results.

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

Views

Author

Wolfdieter Lang, Jan 24 2012

Keywords

Comments

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

Examples

			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

Links

Crossrefs

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

Programs

Formula

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

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.

A206547 Positive odd numbers relatively prime to 21.

Original entry on oeis.org

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 209, 211
Offset: 1

Views

Author

Wolfdieter Lang, Feb 10 2012

Keywords

Comments

These are the positive integers not divisible by 2, 3, or 7.
Numbers coprime to 42. The asymptotic density of this sequence is 2/7. - Amiram Eldar, Oct 23 2020

Links

Crossrefs

Cf. A007775, A204458.

Programs

  • Mathematica
    Select[Range[1,211,2],CoprimeQ[#,21]&] (* Harvey P. Dale, Jul 28 2020 *)

Formula

a(n) = a(n-12) + 42, n>=13.
a(n) = a(n-1) + a(n-12) - a(n-13), n>=13, with a(0)=-1.
a(n) = 2*n-1 + 2*sum(F21[j]*floor((n+(j-1))/12),j=1..12), with F21=[1,2,0,1,0,1,0,1,0,2,1,0], n>=1. For n=0 this becomes -1, but the following o.g.f. has a(0)=0 if it starts with x^0.
O.g.f.: x*(1+x^12+4*x*(1+x^10)+6*x^2*(1+x^8)+2*x^3*(1+x^6)+4*x^4*(1+x^4)+2*x^5*(1+x^2)+4*x^6)/((1-x^12)*(1-x)). The denominator could be factored into cyclotomic polynomials. Compare with the formula contribution from R. J. Mathar in A007775.

A235933 Numbers coprime to 35.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 96, 97, 99
Offset: 1

Views

Author

Oleg P. Kirillov, Jan 17 2014

Keywords

Comments

The asymptotic density of this sequence is 24/35. - Amiram Eldar, Oct 23 2020

Links

Crossrefs

Cf. A160547 (numbers coprime to 31), A229968 (numbers coprime to 33), A204458 (numbers coprime to 34), A007310 (numbers coprime to 36).
Cf. A045572 (numbers not divisible by 5 or 2), A229829 (numbers not divisible by 5 or 3), A047201 (numbers not divisible by 5), A236207 (numbers not divisible by 5 or 11).

Programs

  • Haskell
    a235933 n = a235933_list !! (n-1)
a235933_list = filter ((== 1) . gcd 35) [1..]
-- Reinhard Zumkeller, Mar 27 2014
  • Magma
    [n: n in [1..100] | GCD(n,35) eq 1]; // Bruno Berselli, Mar 27 2014
  • Mathematica
    Select[Range[100], GCD[#, 35] == 1 &] (* Bruno Berselli, Mar 27 2014 *)
  • Sage
    [i for i in range(100) if gcd(i, 35) == 1] # Bruno Berselli, Mar 27 2014

Extensions

Signature corrected from Georg Fischer, Feb 07 2021
Erroneous recurrence removed from Bruno Berselli, Feb 08 2021

A206545 Period length 16: repeat 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1
Offset: 1

Views

Author

Wolfdieter Lang, Feb 09 2012

Keywords

Comments

For general Modd n see a comment on A203571. This sequence gives the Modd 17 residues of the odd numbers not divisible by 17, which are given in A204458.
The underlying periodic sequence with period length 34 is periodic (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 4, 3, 2, 1). This sequence with offset 0 is called P_17 or Modd17.

Examples

			Residue Modd 17 of the positive odd numbers not divisible by 17:
A204458: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29,...
Modd 17: 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11,  9,  7,  5,...

Crossrefs

Cf. A000012 (Modd 3), A084101 (Modd 5), A110551 (Modd 7), A206543 (Modd 11), A206544 (Modd 13).

Programs

  • Mathematica
    PadRight[{},120,Join[Range[1,15,2],Range[15,1,-2]]] (* Harvey P. Dale, Sep 21 2018 *)

Formula

a(n) = A204458(n) (Modd 17) := Modd17(A204458(n)), n>=1, with the periodic sequence Modd17, with period length 34, defined in the comment section.
O.g.f.: x*(1+x^15+3*x*(1+x^13)+5*x^2*(1+x^11)+7*x^3*(1+x^9)+9*x^4*(1+x^7)+11*x^5*(1+x^5)+ 13*x^6*(1+x^3)+15*x^7*(1+x))/(1-x^16) = x*(1+x)^2*(1+x^2)*(1+x^4)/((1+x^8)*(1-x)).
Showing 1-5 of 5 results.

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