A160542 -id:A160542 - 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

A145568 Characteristic function of numbers relatively prime to 11.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Wolfdieter Lang Feb 05 2009

The x-powers appearing in the numerator polynomial of the o.g.f., given below, give the numbers from 0,1,...,10 which survive the sieve of Eratosthenes for multiples of 11, namely 1,2,...10.
Contribution from Reinhard Zumkeller, Nov 30 2009: (Start)
a(n)=A000007(A010880(n)); a(A160542(n))=1; a(A008593(n))=0;
A033443(n) = SUM(a(k)*(n-k): 0<=k<=n). (End)

Index entries for characteristic functions
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

A000035, A011655, A011558, A109720 for coprimality with 2,3,5,7, respectively.

Cf. A168185, A168184, A168182, A168181, A097325, A166486.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

a(n)=gcd(n,11)==1 \\ Charles R Greathouse IV, Jun 28 2015

a(n)=1 if gcd(n,11)=1, else 0. Periodic with period 11: a(n+11)=a(11).
O.g.f.: x*sum(x^k,k=0..9)/(1-x^11).
Completely multiplicative with a(p) = (if p=11 then 0 else 1), p prime. [From Reinhard Zumkeller, Nov 30 2009]
Dirichlet g.f. (1-11^(-s))*zeta(s). - R. J. Mathar, Mar 06 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

A059543 Beatty sequence for log(3).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Mitch Harris, Jan 22 2001

Differs from A160542 at indices n=81, 91, 101, 111, 121, 131, 141, 151, 152, 161 etc. - R. J. Mathar, May 20 2009

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059544.
Cf. A002391 (log(3)).
Cf. A160542.

A059543 := proc(n)
    floor(n*log(3)) ;
end proc:
seq(A059543(n),n=1..100) ; # R. J. Mathar, Jun 26 2023

Floor[Range[100]*Log[3]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(3); for (n = 1, 2000, write("b059543.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*A002391). - Paolo Xausa, Jul 05 2024

A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 130, 131, 132, 140, 141, 142, 143, 150

Offset: 1

Bernard Schott, Jun 27 2022

This definition comes from Patrick De Geest's link.
Other definitions for undulating are present in the OEIS (e.g., A033619, A046075).
When the absolute differences between two adjacent digits are always equal (e.g., 85858), these numbers are called smoothly undulating numbers and form a subsequence (A046075).
The definition includes the trivial 1- and 2-digit undulating numbers.
Subsequence of A043096 where the first different term is A043096(103) = 123 while a(103) = 130.
This sequence first differs from A010784 at a(92) = 101, A010784(92) = 102.
The sequence differs from A160542 (which contains 100). - R. J. Mathar, Aug 05 2022

			111 is not a term here, but A033619(102) = 111.
a(93) = 102, but 102 is not a term of A046075.
Some terms: 5276, 918230, 1053837, 263915847, 3636363636363636.
Are not terms: 1331, 594571652, 824327182.

Patrick De Geest, Smoothly Undulating Palindromic Primes, World of Numbers.

Cf. A033619, A043096, A046075.
Cf. A059168 (subsequence of primes).
Differs from A010784, A241157, A241158.
Cf. A355302, A355303, A355304.

isA355301 := proc(n)
    local dgs,i,back,forw ;
    dgs := convert(n,base,10) ;
    if nops(dgs) < 2 then
        return true;
    end if;
    for i from 2 to nops(dgs)-1 do
        back := op(i,dgs) -op(i-1,dgs) ;
        forw := op(i+1,dgs) -op(i,dgs) ;
        if back*forw >= 0 then
            return false;
        end if ;
    end do:
    back := op(-1,dgs) -op(-2,dgs) ;
    if back = 0 then
        return false;
    end if ;
    return true ;
end proc:
A355301 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA355301(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A355301(n),n=1..110) ; # R. J. Mathar, Aug 05 2022

q[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; Select[Range[0, 100], q] (* Amiram Eldar, Jun 28 2022 *)

isok(m) = if (m<10, return(1)); my(d=digits(m), dd = vector(#d-1, k, sign(d[k+1]-d[k]))); if (#select(x->(x==0), dd), return(0)); my(pdd = vector(#dd-1, k, dd[k+1]*dd[k])); #select(x->(x>0), pdd) == 0; \\ Michel Marcus, Jun 30 2022

A236207 Numbers not divisible by 5 or 11.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 84, 86, 87, 89, 91, 92, 93, 94, 96, 97

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1).

Intersection of A047201 and A160542.

Select[Range[100], Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[100], Or @@ Divisible[#, {5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)

A236217 Numbers not divisible by 3, 5 or 11.

Original entry on oeis.org

1, 2, 4, 7, 8, 13, 14, 16, 17, 19, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119, 122, 124, 127, 128

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 165. The asymptotic density of this sequence is 16/33. - Amiram Eldar, Oct 23 2020

Bruno Berselli, 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, 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).

Intersection of: A160542 and A229829; A047201 and A229968; A001651, A047201 and A160542.

Cf. A236206, A236208.

Select[Range[200], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[200], Or @@ Divisible[#, {3, 5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)
Select[Range[130], CoprimeQ[165, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-80) - a(n-81) for n > 81. - Bruno Berselli, Mar 25 2014

A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

1, 3, 4, 5, 9, 12, 14, 15, 16, 20, 23, 25, 26, 27, 31, 34, 36, 37, 38, 42, 45, 47, 48, 49, 53, 56, 58, 59, 60, 64, 67, 69, 70, 71, 75, 78, 80, 81, 82, 86, 89, 91, 92, 93, 97, 100, 102, 103, 104, 108, 111, 113, 114, 115, 119, 122, 124, 125, 126, 130, 133, 135, 136, 137

Offset: 0

Bruno Berselli, Jan 20 2016

(m^k-1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017401 and k is odd but not multiple of 5 (A045572),
. m is a member of A175885 and k is even but not multiple of 5 (A217562),
. m is a member of A160542 and k is a positive multiple of 10 (A008592),
apart from the trivial case in which k=0.
Also, numbers that are congruent to {1, 3, 4, 5, 9} mod 11. Therefore, the product of two terms belongs to the sequence.
Union of this sequence and A267541 is A160542.
a(n) is prime for n = 1, 3, 10, 14, 17, 21, 24, 27, 30, 33, 40, 44, 47, ...

			From the linear recurrence:
(-A267541) ..., -13, -10, -8, -7, -6, -2, 1, 3, 4, 5, 9, 12, ... (A267755)

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

Cf. A160542, A017329, A267541.

Related sequences (see the first comment): A017401, A160542, A175885.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)));

I:=[1,3,4,5,9,12]; [n le 6 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

gf := (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6): deg := 64: series(gf, x, deg): seq(coeff(%, x, n), n=0..deg-1); # Peter Luschny, Jan 21 2016

CoefficientList[Series[(1 + 2 x + x^2 + x^3 + 4 x^4 + 2 x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 4, 5, 9, 12}, 70]
Select[Range[140], MemberQ[{1, 3, 4, 5, 9}, Mod[#, 11]]&]

Vec((1+2*x+x^2+x^3+4*x^4+2*x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6)
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 21 2016

G.f.: (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267541(n-1).
a(n) = n + 1 + 2*floor(n/5) + 3*floor((n+1)/5) + floor((n+4)/5). - Ridouane Oudra, Sep 06 2023

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A229829 Numbers coprime to 15.

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A145568 Characteristic function of numbers relatively prime to 11.

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A059543 Beatty sequence for log(3).

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A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

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A236207 Numbers not divisible by 5 or 11.

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A236217 Numbers not divisible by 3, 5 or 11.

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A267755 Expansion of (1 + 2*x + x^2 + x^3 + 4*x^4 + 2*x^5)/(1 - x - x^5 + x^6).

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A267755 Expansion of (1 + 2x + x^2 + x^3 + 4x^4 + 2*x^5)/(1 - x - x^5 + x^6).