A047228 -id:A047228 - cp's OEIS frontend

A047241 Numbers that are congruent to {1, 3} mod 6.

1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57, 61, 63, 67, 69, 73, 75, 79, 81, 85, 87, 91, 93, 97, 99, 103, 105, 109, 111, 115, 117, 121, 123, 127, 129, 133, 135, 139, 141, 145, 147, 151, 153, 157, 159, 163, 165, 169, 171, 175, 177, 181, 183

Offset: 1

N. J. A. Sloane

Also the numbers k such that 10^p+k could possibly be prime. - Roderick MacPhee, Nov 20 2011 This statement can be written as follows. If 10^m + k = prime, for any m >= 1, then k is in this sequence. See the pink box comments by Roderick MacPhee from Dec 09 2014. - Wolfdieter Lang, Dec 09 2014
The odd-indexed terms are one more than the arithmetic mean of their neighbors; the even-indexed terms are one less than the arithmetic mean of their neighbors. - Amarnath Murthy, Jul 29 2003
Partial sums are A212959. - Philippe Deléham, Mar 16 2014
12*a(n) is conjectured to be the length of the boundary after n iterations of the hexagon and square expansion shown in the link. The squares and hexagons have side length 1 in some units. The pattern is supposed to become the planar Archimedean net 4.6.12 when n -> infinity. - Kival Ngaokrajang, Nov 30 2014
Positive numbers k for which 1/2 + k/3 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

L. Lovasz, J. Pelikan, K. Vesztergombi, Discrete Mathematics, Springer (2003); 14.4, p. 225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics, Elementary and Beyond, Springer (2003); 14.4, p. 225.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047233, A056970, A007310, A047228, A047261, A047273, A212959.
Subsequence of A186422.
Union of A016921 and A016945. - Wesley Ivan Hurt, Sep 28 2013

a047241 n = a047241_list !! (n-1)
a047241_list = 1 : 3 : map (+ 6) a047241_list
-- Reinhard Zumkeller, Feb 19 2013

seq(3*k-2-((k+1) mod 2), k=1..100); # Wesley Ivan Hurt, Sep 28 2013

Table[{2, 4}, {30}] // Flatten // Prepend[#, 1]& // Accumulate (* Jean-François Alcover, Jun 10 2013 *)
Select[Range[200], MemberQ[{1, 3}, Mod[#, 6]]&] (* or *) LinearRecurrence[{1, 1, -1}, {1, 3, 7}, 70] (* Harvey P. Dale, Oct 01 2013 *)

a(n)=bitor(3*n-3,1) \\ Charles R Greathouse IV, Sep 28 2013

for n in range(1,10**5):print(3*n-2-((n+1)%2)) # Soumil Mandal, Apr 14 2016

From Paul Barry, Sep 04 2003: (Start)
O.g.f.: (1 + 2*x + 3*x^2)/((1 + x)*(1 - x)^2) = (1 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)).
E.g.f.: (6*x + 1)*exp(x)/2 + exp(-x)/2;
a(n) = 3*n - 5/2 - (-1)^n/2. (End)
a(n) = 2*floor((n-1)/2) + 2*n - 1. - Gary Detlefs, Mar 18 2010
a(n) = 6*n - a(n-1) - 8 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 05 2010
a(n) = 3*n - 2 - ((n+1) mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(1)=1, a(2)=3, a(3)=7; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Oct 01 2013
From Benedict W. J. Irwin, Apr 13 2016: (Start)
A005408(a(n)+1) = A016813(A001651(n)),
A007310(a(n)) = A005408(A087444(n)-1),
A007310(A005408(a(n)+1)) = A017533(A001651(n)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Dec 11 2021

Formula corrected by Bruno Berselli, Jun 24 2010

A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

N. J. A. Sloane

Complement of A047235. - Reinhard Zumkeller, Oct 01 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004524, A007310, A047228, A047235, A047241, A047261, A093719, A096981.
First differences of A281026.
See A301729 for an essentially identical sequence.

a047273 n = a047273_list !! (n-1)
a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list
-- Reinhard Zumkeller, Feb 19 2013

[(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 20 2016

seq(2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, n = 0..69); # Gary Detlefs, Mar 19 2010

LinearRecurrence[{2,-2,2,-1},{0,1,3,5},80] (* Harvey P. Dale, Jan 04 2015 *)

PARI
```
a(n)=n+(n+1)\4+(n+2)\4
```

[(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # Zerinvary Lajos, Mar 09 2009

G.f.: x*(1+x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^2)*(1-x^3)/((1-x)^3*(1-x^4)).
a(n) = n + A004524(n+1) = -a(-n) for all n in Z.
Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
A093719(a(n)) = 1. - Reinhard Zumkeller, Oct 01 2008
a(n) = 2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, with offset 0..a(0)=0. - Gary Detlefs, Mar 19 2010
a(n) = (3*n-3+cos(Pi*n/2))/2. - R. J. Mathar, Oct 08 2010
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.
a(n) = (6*n-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)
Euler transform of length 4 sequence [3, -1, -1, 1]. - Michael Somos, Jun 24 2017
Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/2. - Amiram Eldar, Dec 16 2021
E.g.f.: (2 + 3*exp(x)*(x - 1) + cos(x))/2. - Stefano Spezia, Jul 26 2024

A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101, 103, 104

Offset: 1

N. J. A. Sloane

Each element is coprime to preceding two elements. - Amarnath Murthy, Jun 12 2001

The sequence is the interleaving of A047241 with A016789. - Guenther Schrack, Feb 16 2019

			After 21 and 23 the next term is 25 as 24 has a common divisor with 21.

Guenther Schrack, Table of n, a(n) for n = 1..10012
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A007310, A016789, A047228, A047237, A047241, A047251, A047261, A047273, A062062, A062063.

Complement: A047233

a047255 n = a047255_list !! (n-1)
a047255_list = 1 : 2 : 3 : 5 : map (+ 6) a047255_list
-- Reinhard Zumkeller, Jan 17 2014

[n : n in [0..100] | n mod 6 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, May 21 2016

A047255:=n->(6*n-4+I^(1-n)+I^(n-1))/4: seq(A047255(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Select[Range[100], MemberQ[{1, 2, 3, 5}, Mod[#, 6]] &]
LinearRecurrence[{2,-2,2,-1},{1,2,3,5},100] (* Harvey P. Dale, May 14 2020 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,2,-2,2]^(n-1)*[1;2;3;5])[1,1] \\ Charles R Greathouse IV, Feb 11 2017

a=(x*(1+x^2+x^3)/((1+x^2)*(1-x)^2)).series(x, 80).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

{k | k == 1, 2, 3, 5 (mod 6)}.
G.f.: x*(1 + x^2 + x^3) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4), for n>4.
a(n) = (6*n - 4 + i^(1-n) + i^(n-1))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047241(n). (End)
E.g.f.: (2 + sin(x) + (3*x - 2)*exp(x))/2. - Ilya Gutkovskiy, May 21 2016
a(1-n) = - A047251(n). - Wesley Ivan Hurt, May 21 2016
From Guenther Schrack, Feb 16 2019: (Start)
a(n) = (6*n - 4 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=1, a(2)=2, a(3)=3, a(4)=5, for n > 4.
a(n) = A047237(n) + 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 + log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021
a(n) = 2*n - 1 - floor(n/2) + floor(n/4) - floor((n+1)/4). - Ridouane Oudra, Feb 21 2023

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001

A047261 Numbers that are congruent to {2, 4, 5} mod 6.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 14, 16, 17, 20, 22, 23, 26, 28, 29, 32, 34, 35, 38, 40, 41, 44, 46, 47, 50, 52, 53, 56, 58, 59, 62, 64, 65, 68, 70, 71, 74, 76, 77, 80, 82, 83, 86, 88, 89, 92, 94, 95, 98, 100, 101, 104, 106, 107, 110, 112, 113, 116, 118, 119, 122, 124

Offset: 1

N. J. A. Sloane

If B and C are terms in the sequence then 2*B*C is a term. B (resp. C) is a term iff B (resp. C) mod 6 = 2, 4 or 5. It follows that (2*B*C) mod 6 = (2*(B mod 6)*(C mod 6)) mod 6 = 2 or 4 and therefore 2*B*C is a term. Examples: for B=16 and C=29, 2*16*29 = 928 is a term: (2*B*C) mod 6 = (2*16*29) mod 6 = 4; (2*2*2) mod 6 = 2. - Jerzy R Borysowicz, May 24 2018

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

Cf. A047242 (complement).

Cf. A007310, A047228, A047241, A047273, A056970, A214090.

a047261 n = a047261_list !! n
a047261_list = 2 : 4 : 5 : map (+ 6) a047261_list
-- Reinhard Zumkeller, Feb 19 2013, Jul 06 2012

[n : n in [0..150] | n mod 6 in [2, 4, 5]]; // Wesley Ivan Hurt, Jun 14 2016

A047261:=n->(6*n-1-2*cos(2*n*Pi/3))/3: seq(A047261(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

CoefficientList[Series[(1 + x)*(x^2 + 2)/((1 + x + x^2)*(x - 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Aug 16 2014 *)
Select[ Range@ 125, MemberQ[{2, 4, 5}, Mod[#, 6]] &] (* or *)
LinearRecurrence[{1, 0, 1, -1}, {2, 4, 5, 8}, 62] (* Robert G. Wilson v, Jun 13 2018 *)

G.f.: x*(1+x)*(x^2+2) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
A214090(a(n)) = 1. - Reinhard Zumkeller, Jul 06 2012
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (6*n - 1 - 2*cos(2*n*Pi/3))/3.
a(3k) = 6k-1, a(3k-1) = 6k-2, a(3k-2) = 6k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2+sqrt(3))/(2*sqrt(3)) + log(2)/3. - Amiram Eldar, Dec 16 2021
E.g.f.: (3 + exp(x)*(6*x - 1) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 26 2024

More terms from Wesley Ivan Hurt, Aug 16 2014

A083039 Number of divisors of n that are <= 3.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3

Offset: 1

Daniele A. Gewurz (gewurz(AT)mat.uniroma1.it) and Francesca Merola (merola(AT)mat.uniroma1.it), May 06 2003

Periodic of period 6. Parker vector of the wreath product of S_3 and S, the symmetric group of a countable set.

			The divisors of 6 are 1, 2, 3 and 6. Of those divisors, 1, 2 and 3 are <= 3. That's three divisors, therefore, a(6) = 3. - _David A. Corneth_, Sep 30 2017

Antti Karttunen, Table of n, a(n) for n = 1..12000
D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6
M. D. Hirschhorn, J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A000005, A007310, A008588, A047228, A083040, A138553.

LinearRecurrence[{-1, 0, 1, 1},{1, 2, 2, 2},90] (* Ray Chandler, Aug 26 2015 *)

PARI
```
a(n)=[3,1,2,2,2,1][n%6+1];
```

G.f.: x/(1-x) + x^2/(1-x^2) + x^3/(1-x^3).
a(n) = a(n-6) = a(-n).
a(n) = 11/6 - (1/2)*(-1)^n - (1/3)*cos(2*Pi*n/3) - (1/3)*3^(1/2)*sin(2*Pi*n/3). - Richard Choulet, Dec 12 2008
a(n) = Sum_{k=1..1} cos(n*(k - 1)/1*2*Pi)/1 + Sum_{k=1..2} cos(n*(k - 1)/2*2*Pi)/2 + Sum_{k=1..3} cos(n*(k - 1)/3*2*Pi)/3. - Mats Granvik, Sep 09 2012
a(n) = log_2(gcd(n,2) + gcd(n,6)). - Gary Detlefs, Feb 15 2014
a(n) = Sum_{d|n, d<=3} 1. - Wesley Ivan Hurt, Oct 30 2023

A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 5, 4, 7, 6, 11, 9, 15, 14, 22, 20, 31, 29, 43, 41, 58, 57, 80, 78, 106, 107, 142, 143, 188, 191, 247, 253, 321, 332, 418, 432, 537, 561, 690, 721, 880, 924, 1118, 1178, 1412, 1493, 1781, 1884, 2231, 2370, 2789, 2965, 3472, 3698, 4309, 4596

Offset: 0

Vladeta Jovovic, Aug 23 2004

Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - Emeric Deutsch, Feb 16 2006

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			a(8)=5 because we have [8],[44],[422],[332] and [2222].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A070047.

Cf. A047228, A056970, A096981, A098884.

a097451 n = p a047228_list n where
   p _  0         = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)),j=0..15): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..67); # Emeric Deutsch, Feb 16 2006

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 24 2013 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 24 2013 */

Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].
G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - Emeric Deutsch, Feb 16 2006
Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 24 2013
Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Sep 24 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015

More terms from Emeric Deutsch, Feb 16 2006

A374445 Lexicographically earliest sequence of distinct positive integers such that any pair of consecutive terms are coprime whereas the squarefree kernel of their product is primorial.

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 15, 14, 45, 16, 27, 10, 21, 20, 63, 40, 81, 32, 75, 28, 135, 56, 165, 98, 225, 64, 105, 22, 315, 44, 525, 88, 735, 128, 243, 50, 147, 80, 189, 100, 231, 130, 693, 160, 441, 110, 273, 220, 567, 200, 729, 70, 33, 140, 99, 280, 297, 350, 363

Offset: 1

David James Sycamore, Jul 08 2024

nonn

In other words rad(a(n-2)*a(n-1)) is a term in A002110 whereas a(n-2) and a(n-1) share no common divisor > 1. Every term > a(1) = 1 is divisible by 2 or by 3 but not by both, and all terms other than 1,2,3 are composite.

{a(n); n >= 2} is conjectured to be a permutation of A047228.

			The sequence starts with a(1) = 1, a(2) = 2 since (1,2) = 1 and 1*2 = A002110(1).
a(3) = 3 since (2,3) = 1 and 2*3 = 6 = A002110(2).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10000, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, with magenta representing powerful numbers that are not prime powers.

Cf. A002110, A007947, A047228, A374351.

nn = 540; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = a[2]; u = 3;
f[x_] := f[x] = Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Monitor[Do[k = u;
  While[Or[! CoprimeQ[j, k], c[k], ! f[j*k]], k++];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}], n];
Array[a, nn] (* Michael De Vlieger, Jul 16 2024 *)

PARI
```
\\ See Links section.
```

More terms from Rémy Sigrist, Jul 11 2024

A365117 a(1) = 1. Thereafter a(n) is the least novel multiple m of the smallest prime which does not divide a(n-1) and such that m is coprime to a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 15, 14, 27, 10, 21, 16, 33, 20, 39, 22, 45, 26, 51, 28, 57, 32, 63, 34, 69, 38, 75, 44, 81, 40, 87, 46, 93, 50, 99, 52, 105, 58, 111, 56, 117, 62, 123, 64, 129, 68, 135, 74, 141, 70, 153, 76, 147, 80, 159, 82, 165, 86, 171, 88, 177, 92, 183, 94

Offset: 1

David James Sycamore, Aug 22 2023

nonn

The Name is as for A351495, but with an extra constraint: namely that a(n) is prime to a(n-1).

{a(n)}; n > 1 is a permutation of A047228.

			a(2) = 2 because 2 is the smallest prime which does not divide 1, and 2 is prime to 1.
3(3) = 3 because 3 is the smallest prime which does not divide 2, and 3 is prime to 2.
a(4) = 4 since it is the second multiple of 2, the smallest prime that does not divide 3, and 4 is prime to 3.
a(5) = 9 since it is the least novel multiple of 3, the smallest prime that does not divide 4, and is the least such number prime to 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..256, showing a(n) mod 6 = 3 in red, a(n) mod 6 = 2 in large dark blue, and a(n) mod 6 = 4 in green dots. Demonstrates 2 trajectories, an early trajectory pertaining to 3 (mod 6) and a late trajectory pertaining to +-2 (mod 6).
Michael De Vlieger, 32 X 32 table of a(n), n = 1..1024, listed in rows, with a color code according to residue (mod 6). Blue = 2 (mod 6), red = 3 (mod 6), light green = 4 (mod 6), black = 0, 1, or 5 mod 6.
Michael De Vlieger, 30 X 30 table of a(n), n = 1..900, listed in rows, with a color code as follows: black = empty product, red = prime, gold = composite prime power, light green = squarefree semiprime, green = squarefree number with more than 2 prime factors, purple = powerful number (in A001694) that is not a prime power, blue = other numbers that are neither squarefree nor prime powers.
Michael De Vlieger, 210 X 210 table of a(n), n = 1..44100, listed in rows with color code as immediately above, demonstrating the aggregate pattern demonstrated in the sequence. Certain residues (mod 210) seem to be devoid of squarefree numbers, perhaps since they are a product 9*m.

Cf. A047228, A351495.

nn = 12; m[] := 1; a[1] = j = 1; m[1] = 2; c[1] = True; c[] := False; f[x_] := Block[{q}, q = 2; While[! CoprimeQ[q, x], q = NextPrime[q]]; q]; Do[{k = m[#]; While[Or[! CoprimeQ[j, k], c[# k]], k++]; If[k == m[#], While[c[m[#] #], m[#]++]]; Set[{a[n], j, c[k #]}, {k #, k #, True}]} &@ f[j], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 22 2023 *)

first(n) = {my(res = vector(n)); sofar = Set([1..4]); for(i = 1, 4, res[i] = i); for(i = 5, n, res[i] = nxt(res[i-1])); res}
nxt(n) = {my(start, step); if(n % 2 == 0, start = 3; step = 6, start = 2; step = [2,4]); forstep(i = start, oo, step, s = Set(i); if(gcd(i, n) == 1 && #setminus(s, sofar) == 1, sofar = setunion(sofar, s); return(i)))} \\ David A. Corneth, Aug 22 2023

From Michael De Vlieger, Aug 22 2023: (Start)
a(2n) == +-2 (mod 6).
a(2n+1) == 3 (mod 6), n > 0. (End)

More terms from David A. Corneth, Aug 22 2023

A374277 Numbers k divisible by exactly one of the prime factors of 30.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 74, 76, 81, 82, 85, 86, 87, 88, 92, 93, 94, 95, 98, 99, 104, 106, 111, 112, 115, 116, 117, 118, 122, 123, 124, 125, 128, 129, 134

Offset: 1

Michael De Vlieger, Jul 26 2024

Numbers k congruent to r (mod 30), where r is in {2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28}, residues r = p^m mod 30 and r = (30 - p^m) mod 30.

The asymptotic density of this sequence is 7/15. - Amiram Eldar, Jul 26 2024

			8 is in this sequence since it is even and a multiple of neither 3 nor 5.
10 is not in this sequence since 10 = 2*5; both 2 and 5 divide 30.
14 is in this sequence since it is even and a multiple of neither 3 nor 5, etc.

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

Cf. A047228, A051037, A306044.

s = Prime@ Range[3]; k = Times @@ s; r = Union[#, k - #] &@ Flatten@ Map[PowerRange[#, k, #] &, s]; m = Length[r]; Array[k*#1 + r[[1 + #2]] & @@ QuotientRemainder[# - 1, m] &, 60]

is(k) = isprime(gcd(k, 30)); \\ Amiram Eldar, Jul 26 2024

Intersection of this sequence and 5-smooth numbers (A051037) is A306044 \ {1}.

cp's OEIS Frontend

A047241 Numbers that are congruent to {1, 3} mod 6.

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A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

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A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

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A047261 Numbers that are congruent to {2, 4, 5} mod 6.

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A083039 Number of divisors of n that are <= 3.

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A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

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