A047246 -id:A047246 - cp's OEIS frontend

A045331 Primes congruent to {1, 2, 3} mod 6; or, -3 is a square mod p.

2, 3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane

-3 is a quadratic residue mod a prime p iff p is in this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Apart from initial term, same as A007645; apart from initial two terms, same as A002476.
Cf. A002313, A033203, A038873, A038874, A057128.
Subsequence of A047246.

a045331 n = a045331_list !! (n-1)
a045331_list = filter ((< 4) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

[p: p in PrimesUpTo(700) | p mod 6 in [1, 2, 3]]; // Vincenzo Librandi, Aug 08 2012

Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,6]]&]  (* Harvey P. Dale, Mar 31 2011 *)
Join[{2,3},Select[Range[7,10^3,6],PrimeQ]] (* Zak Seidov, May 20 2011 *)

select(n->n%6<5,primes(100)) \\ Charles R Greathouse IV, May 20 2011

More terms from Henry Bottomley, Aug 10 2000

A047257 Numbers that are congruent to {4, 5} mod 6.

Original entry on oeis.org

4, 5, 10, 11, 16, 17, 22, 23, 28, 29, 34, 35, 40, 41, 46, 47, 52, 53, 58, 59, 64, 65, 70, 71, 76, 77, 82, 83, 88, 89, 94, 95, 100, 101, 106, 107, 112, 113, 118, 119, 124, 125, 130, 131, 136, 137, 142, 143, 148, 149

Offset: 1

N. J. A. Sloane

Equivalently, numbers m such that 2^m - m is divisible by 3. Indeed, for every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 3. - Bernard Schott, Dec 10 2021

Numbers k for which A276076(k) and A276086(k) are multiples of nine. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

David Lovler, Table of n, a(n) for n = 1..1000
The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index to sequences related to Olympiads.

Cf. A000325.
Similar with: A299174 (p = 2), this sequence (p = 3), A349767 (p = 5).
Cf. also A047227, A047235, A047246, A047247, A047270.
Cf. A007623, A049345, A276076, A276086.

Select[Range@ 150, 4 <= Mod[#, 6] <= 5 &] (* Michael De Vlieger, Mar 20 2015 *)
LinearRecurrence[{1,1,-1},{4,5,10},50] (* Harvey P. Dale, Oct 16 2017 *)

A047257(n):=4 + 6*floor(n/2) + mod(n,2)$ akelist(A047257(n),n,0,40); /* Martin Ettl, Oct 24 2012 */

a(n) = 3*n - (-1)^n \\ David Lovler, Aug 25 2022

a(n) = 4 + 6*floor(n/2) + n mod 2.
a(n) = 6*n-a(n-1)-3, with a(1)=4. - Vincenzo Librandi, Aug 05 2010
G.f.: ( x*(4+x+x^2) ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = 3*n - (-1)^n. - Wesley Ivan Hurt, Mar 20 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) - log(2)/3. - Amiram Eldar, Dec 14 2021
E.g.f.: 1 + 3*x*exp(x) - exp(-x). - David Lovler, Aug 25 2022

A047247 Numbers that are congruent to {2, 3, 4, 5} (mod 6).

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 62, 63, 64, 65, 68, 69, 70, 71, 74, 75, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 95, 98, 99

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047235 with A047270. - Guenther Schrack, Feb 10 2019

Numbers k for which A276076(k) and A276086(k) are multiples of three. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

Guenther Schrack, Table of n, a(n) for n = 1..10015
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for two-way infinite sequences.

Cf. A047227, A047235, A047246, A047270, A047257.
Cf. A007623, A049345, A276076, A276086.
Complement: A047225.

[n : n in [0..100] | n mod 6 in [2, 3, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

A047247:=n->(6*n-1-I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/4: seq(A047247(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-1-I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,4,5,8},70] (* Harvey P. Dale, May 25 2024 *)

my(x='x+O('x^70)); Vec(x*(2+x+x^2+x^3+x^4)/((1-x)*(1-x^4))) \\ G. C. Greubel, Feb 16 2019

a=(x*(2+x+x^2+x^3+x^4)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x*(2+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(1-x)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 1 - i^(2*n) - (1-i)*i^(-n) - (1+i)*i^n)/4 where i = sqrt(-1).
a(2*n) = A047270(n), a(2*n-1) = A047235(n).
a(n) = A047227(n) + 1, a(1-n) = - A047227(n). (End)
From Guenther Schrack, Feb 10 2019: (Start)
a(n) = (6*n - 1 - (-1)^n -2*(-1)^(n*(n+1)/2))/4.
a(n) = a(n-4) + 6, a(1)=2, a(2)=3, a(3)=4, a(4)=5, for n > 4.
a(n) = A047227(n) + 1. a(n) = A047246(n) + 2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 - 2*log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021

More terms from Wesley Ivan Hurt, May 21 2016

A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98

Offset: 1

N. J. A. Sloane

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

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

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.

Complement of A047264. Equals A203016 divided by 3.

[n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013

A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

From Johannes W. Meijer, Jul 07 2011: (Start)
a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = A047246(n) + 1.
a(n+2) - a(n+1) = A093148(n) for n>0.
a(1-n) = - A047247(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

Original entry on oeis.org

0, 3, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 60, 63, 64, 65, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 99

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047233 with A047270. - Guenther Schrack, Feb 15 2019

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047233, A047246, A047270.

Complement: A047239.

[n : n in [0..150] | n mod 6 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 02 2016

A047250:=n->(6*n-3+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/4: seq(A047250(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[0,100], MemberQ[{0,3,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,3,4,5,6}, 60] (* Harvey P. Dale, Apr 01 2013 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 3 + i^(2*n) - (1+i)*i^(-n) - (1-i)*i^n)/4 where i=sqrt(-1).
a(2*k) = A047270(k), a(2*k-1) = A047233(k). (End)
E.g.f.: (2 - sin(x) - cos(x) + (3*x - 2)*sinh(x) + (3*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
From Guenther Schrack, Feb 15 2019: (Start)
a(n) = (6*n - 3 + (-1)^n - 2*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=3, a(3)=4, a(4)=5, for n > 4.
a(-n) = -A047246(n+2). (End)
Sum_{n>=2} (-1)^n/a(n) = 2*log(2)/3 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 17 2021

A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 90

Offset: 0

Bruno Berselli, May 03 2016

Also, numbers that are congruent to {0..6} mod 9.

The initial terms coincide with those of A037475 and A039111. First disagreement is after 60 (index 48): a(49) = 63, A037475(49) = 81 and A039111(50) = 71.

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

Cf. A248375: f(9,n).

Cf. similar sequences with the formula f(k, f(k-1, n)): A008585 (k=3), A042948 (k=4), A047217 (k=5), A047246 (k=6), A047337 (k=7), A047602 (k=8), this sequence (k=9), A272576 (k=10).

k:=9; f:=func; [f(k,f(k-1,n)): n in [0..70]];

f := (k, m) -> floor(m*k/(k-1)):
a := n -> f(9, f(8, n)):
seq(a(n), n = 0..70); # Peter Luschny, May 03 2016

f[k_, m_] := Floor[m*k/(k-1)];
a[n_] := f[9, f[8, n]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 09 2016 *)
CoefficientList[Series[x (1 + x + x^2 + x^3 + x^4 + x^5 + 3 x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 70}], x] (* or *)
Table[(63 n - 12 - 12 Mod[n, 7] + 2 Mod[-n - 1, 7])/49, {n, 0, 70}] (* Michael De Vlieger, Dec 25 2016 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,9},90] (* Harvey P. Dale, May 08 2018 *)

f = lambda k, m: floor(m*k/(k-1))
a = lambda n: f(9, f(8, n))
[a(n) for n in range(71)] # Peter Luschny, May 03 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + 3*x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8).
a(n) = (63*n - 12 - 12*(n mod 7) + 2*((-n-1) mod 7))/49. - Wesley Ivan Hurt, Dec 25 2016

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A045331 Primes congruent to {1, 2, 3} mod 6; or, -3 is a square mod p.

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A047257 Numbers that are congruent to {4, 5} mod 6.

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A047247 Numbers that are congruent to {2, 3, 4, 5} (mod 6).

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A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

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A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

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A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

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