A047235 -id:A047235 - cp's OEIS frontend

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

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

A020832 Decimal expansion of 1/sqrt(75).

Original entry on oeis.org

1, 1, 5, 4, 7, 0, 0, 5, 3, 8, 3, 7, 9, 2, 5, 1, 5, 2, 9, 0, 1, 8, 2, 9, 7, 5, 6, 1, 0, 0, 3, 9, 1, 4, 9, 1, 1, 2, 9, 5, 2, 0, 3, 5, 0, 2, 5, 4, 0, 2, 5, 3, 7, 5, 2, 0, 3, 7, 2, 0, 4, 6, 5, 2, 9, 6, 7, 9, 5, 5, 3, 4, 4, 6, 0, 5, 8, 6, 6, 6, 9, 1, 3, 8, 7, 4, 3, 0, 7, 9, 1, 1, 7, 1, 4, 9, 9, 0, 5

Offset: 0

N. J. A. Sloane

Multiplied by 10 this is 2/sqrt(3). - Alonso del Arte, Apr 30 2012
2/sqrt(3) is Hermite's constant gamma_2. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
2/sqrt(3) is the Lorentz factor for an object traveling at half the speed of light. - Sean Stroud, May 05 2019

			0.1154700538379251529...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 62.
Yining Hu, Patterns in numbers and infinite sums and products, arXiv:1506.00151 [math.NT], 2015.
Samuel G. Moreno and Esther M. García, New Infinite Products of Cosines and Viète-Like Formulae, Mathematics Magazine, vol. 86, no. 1, 2013, pp. 15-25. See formula for 2/sqrt(3) page 15.
Index entries for algebraic numbers, degree 2

Cf. A010153 (continued fraction, but missing the initial 0), A047235.

RealDigits[1/Sqrt[75], 10, 100][[1]] (* Alonso del Arte, Apr 30 2012 *)

75^-.5 \\ Charles R Greathouse IV, Mar 31 2016

(csc(Pi/3))/10, where csc is the cosecant function. - Alonso del Arte, Apr 30 2012
Product_{n>=1} ((3*n+1)/(3*n+2))^((-1)^n), with offset 1. (see Hu link). - Michel Marcus, Jun 02 2015
From Amiram Eldar, Aug 02 2020: (Start)
2/sqrt(3) = Sum_{k>=0} binomial(2*k,k)/16^k.
2/sqrt(3) = 1 + Sum_{k>=1} (2*k-1)!!/((2*k)!! * 2^(2*k)). (End)
2/sqrt(3) = Product_{k>=1} (1 - (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024

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

A120325 Period 6: repeat [0, 0, 1, 0, 1, 0].

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 21 2006

Dirichlet series for the principal character mod 6: L(s,chi) = Sum_{n>=1} a(n+3)/n^s = (1 + 1/6^s - 1/2^s - 1/3^s) Riemann-zeta(s), e.g., L(2,chi) = A100044, L(4,chi) = 5*Pi^4/486, L(6,chi) = 91*Pi^6/87480. See Jolley eq (313) and arXiv:1008.2547 L(m=6,r=1,s). - R. J. Mathar, Jul 31 2010

			a(0) = (1/3)*(sin(0) + sin(0))^2 = 0.
a(1) = (1/3)*(sin(Pi/6) + sin(7*Pi/6))^2 = (1/3)*(1/2 - 1/2)^2 = 0.
a(2) = (1/3)*(sin(Pi/3) + sin(7*Pi/3))^2 = (1/3)*((sqrt(3))/2 + (sqrt(3))/2)^2 = 1.
a(3) = (1/3)*(sin(Pi/2) + sin(7*Pi/2))^2 = (1/3)*(1 - 1)^2 = 0.
a(4) = (1/3)*(sin(2*Pi/3) + sin(14*Pi/3))^2 = (1/3)*((sqrt(3))/2 + (sqrt(3))/2)^2 = 1.
a(5) = (1/3)*(sin(5*Pi/6) + sin(35*Pi/6))^2 = (1/3)*(1/2 - 1/2)^2 = 0.

L. B. W. Jolley, Summation of Series, Dover (1961).

Antti Karttunen, Table of n, a(n) for n = 0..100000
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Characteristic function of A047235. One's complement of A093719.

Cf. A100044, A059841, A276086, A358841, A358842, A358754, A358755.

[(n+3)^2 mod (2+((n+1) mod 2)) : n in [0..100]]; // Wesley Ivan Hurt, Oct 31 2014

P:=proc(n)local i,j; for i from 0 by 1 to n do j:=1/3*(sin(i*Pi/6)+sin(7*i*Pi/6))^2; print(j); od; end: P(20);
seq(abs(numtheory[jacobi](n,6)),n=3..150) ; # R. J. Mathar, Jul 31 2010

Table[Mod[(n + 3)^2, (5 + (-1)^n)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 31 2014 *)
PadRight[{},120,{0,0,1,0,1,0}] (* Harvey P. Dale, Oct 05 2016 *)

A120325(n) = ((n%3)&&!(n%2)); \\ Antti Karttunen, Dec 03 2022

def A120325(n): return int(not (n+3) % 6 & 3 ^ 1) # Chai Wah Wu, May 25 2022

a(n) = (1/3)*(sin(n*Pi/6) + sin(7*n*Pi/6))^2.
From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^2*(1+x^2)/((1+x)*(1-x)*(1+x+x^2)*(1-x+x^2)).
a(n+6) = a(n). (End)
a(n) = ((n+3)*Fibonacci(n+3)) mod 2. - Gary Detlefs, Dec 13 2010
a(n) = 0 if n mod 6 = 0, otherwise a(n) = n mod 2 + (-1)^n. - Gary Detlefs, Dec 13 2010
a(n) = (n+3)^2 mod (5+(-1)^n)/2. - Wesley Ivan Hurt, Oct 31 2014
a(n) = sin(n*Pi/3)^2*(2-4*cos(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 19 2016
E.g.f.: 2*(cosh(x) - cos(sqrt(3)*x/2)*cosh(x/2))/3. - Ilya Gutkovskiy, Jun 20 2016
a(n) = sign((n-3) mod 2) * sign((n-3) mod 3). - Wesley Ivan Hurt, Feb 04 2022
From Antti Karttunen, Dec 03 2022: (Start)
a(n) = 1 - A093719(n).
a(n) = [A276086(n) == 3 (mod 6)], where [ ] is the Iverson bracket.
a(n) = A059841(n) - A358841(n) - A358842(n).
For n >= 1, a(n) = A059841(n) - A358754(n) - A358755(n).
(End)

Data section extended up to a(120) by Antti Karttunen, Dec 03 2022

A358840 Primorial base exp-function reduced modulo 6.

Original entry on oeis.org

1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 1, 2, 3, 0, 3, 0, 5, 4, 3, 0, 3, 0, 1, 2, 3, 0

Offset: 0

Antti Karttunen, Dec 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A010875, A276086, A328632 (positions of 1's), A358841 (their characteristic function), A047235 (positions of 3's), A120325 (their char. fun), A358843 (positions of 5's), A358842 (their char. fun).

Cf. also A353486, A358850.

A358840(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m%6); };

a(n) = A010875(A276086(n)) = A276086(n) mod 6.

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

A328588 Numbers n for which A257993(A276086(A276086(n))) is larger than A257993(n), where A276086 converts the primorial base expansion of n into its prime product form, and A257993 returns the index of the least prime not present in its argument.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212, 214, 218, 220, 224, 226, 230, 232, 236, 238, 240, 242

Offset: 1

Antti Karttunen, Oct 21 2019

nonn

Numbers n for which A328578(n) > A257993(n).

A047235 (numbers that are congruent to {2, 4} mod 6, thus even numbers that are not multiples of 3, with A257993(n) = 1) is a subsequence, because in primorial base (A049345) such numbers end with digits "10" or "20". A276086 will convert such a number to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^{1,2} * 2^0 (an odd multiple of three, thus of the form 6k+3) which in primorial base will end with digits "11", thus on the second iteration A276086 will convert that to a number of the form p_k^e_k * ... * 7^b * 5^a * 3^1 * 2^1, with the least missing prime having an index (A257993) at least 3, which is larger than the original 1. Thus all terms of A047235 are included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Cf. A049345, A047235, A257993, A276086, A328578, A328585, A328586, A328587.
Union of A047235 (terms of the form 6k+2 and 6k+4) and A328589 (gives the terms that are multiples of 6).
Positions of positive terms in A328590.
Differs from A047235 for the first time at n=81, with a(81) = 240, a term not present in A047235.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328588(n) = (A328578(n) > A257993(n));

A102039 a(n) = a(n-1) + last digit of a(n-1), starting at 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 22, 24, 28, 36, 42, 44, 48, 56, 62, 64, 68, 76, 82, 84, 88, 96, 102, 104, 108, 116, 122, 124, 128, 136, 142, 144, 148, 156, 162, 164, 168, 176, 182, 184, 188, 196, 202, 204, 208, 216, 222, 224, 228, 236, 242, 244, 248, 256, 262, 264, 268, 276, 282

Offset: 1

Samantha Stones (devilsdaughter2000(AT)hotmail.com), Dec 25 2004

Sequence A001651 is the "base 3" version. In base 4 this rule leads to (1,2,4,4,4...), in base 5 to (1,2,4,8,11,12,14,18,21,22,24,28...) = A235700. - M. F. Hasler, Jan 14 2014

This and the following sequences (none of which is "base"!) could all be defined by a(1) = 1 and a(n+1) = a(n) + (a(n) mod b) with different values of b: A001651 (b=3), A235700 (b=5), A047235 (b=6), A047350 (b=7), A007612 (b=9). Using b=4 or b=8 yields a constant sequence from that term on. - M. F. Hasler, Jan 15 2014

			28 + 8 = 36, 36 + 6 = 42.

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

Apart from initial term, same as A002081.

LinearRecurrence[{2,-2,2,-1},{1,2,4,8,16},60] (* Harvey P. Dale, Jul 02 2022 *)

print1(a=1);for(i=1,99,print1(","a+=a%10)) \\ M. F. Hasler, Jan 14 2014

Vec(x*(5*x^4+2*x^3+2*x^2+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Sep 20 2014

a(n) = if(n==1, 1, (-10-(1-I/2)*(-I)^n-(1+I/2)*I^n+5*n)) \\ Colin Barker, Oct 18 2015

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>5. G.f.: x*(5*x^4+2*x^3+2*x^2+1) / ((x-1)^2*(x^2+1)). - Colin Barker, Sep 20 2014

a(n) = (-10 - (1-i/2)*(-i)^n - (1+i/2)*i^n + 5*n) for n>1, where i = sqrt(-1). - Colin Barker, Oct 18 2015

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

A066284 a(n) = A066135(4*n).

Original entry on oeis.org

34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 386, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194, 34, 84, 34, 84, 34, 228, 34, 84, 34, 84, 34, 194

Offset: 1

Labos Elemer, Dec 11 2001

nonn

a(n) <= 2p, where p = A002586(4n) is the least prime factor of (1 + 16^n). (See the Comment in A066135.) - Jonathan Sondow, Nov 23 2012

			First 3 terms correspond to entries of other sequences as follows: a(1)=A046763(2), a(2)=A055712(2), a(3)=A055716(2).
From _Michael De Vlieger_, Jul 17 2017: (Start)
First position of values, with observations pertaining to values for 1 <= n <= 3000:
    Value   Position   Observations:
    --------------------------------
       34     1        All odd.
       84     2        In A047235.
      194     6        In A017593.
      228    12
      386    36
     1282    72
     1538   144
     3084   288
   147468   576
     1956   864
  1046532  1152
    24578  2304
     3252  2880
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A066135, A046761, A046762, A046763, A055709-A055717, A007691, A001159.

Table[m = 2; While[Mod[DivisorSigma[4 n, m], m] > 0, m++]; m, {n, 66}] (* Michael De Vlieger, Jul 17 2017 *)

a(n) = {n *= 4; my(m = 2); while (sigma(m, n) % m, m++); m;} \\ Michel Marcus, Oct 02 2016

a(n) = Min{x : sigma_4n(x) mod x = 0, x > 1}

cp's OEIS Frontend

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