A213243 -id:A213243 - cp's OEIS frontend

A083420 a(n) = 2*4^n - 1.

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Paul Barry, Apr 29 2003

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.

a083420 = subtract 1 . (* 2) . (4 ^)  -- Reinhard Zumkeller, Dec 22 2015

[2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015

seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011

2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)

a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015

def A083420(n): return (1<<(n<<1|1))-1 # Chai Wah Wu, Mar 10 2025

G.f.: (1+2*x)/((1-x)*(1-4*x)).
E.g.f.: 2*exp(4*x)-exp(x).
With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003
From Benoit Cloitre, Jun 18 2004: (Start)
a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006
a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006
a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010
a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011
a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012
a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014
a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019
a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020
a(n) = numerator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022

A213245 Number of nonzero elements in GF(2^n) that are 7th powers.

Original entry on oeis.org

1, 3, 1, 15, 31, 9, 127, 255, 73, 1023, 2047, 585, 8191, 16383, 4681, 65535, 131071, 37449, 524287, 1048575, 299593, 4194303, 8388607, 2396745, 33554431, 67108863, 19173961, 268435455, 536870911, 153391689, 2147483647, 4294967295, 1227133513, 17179869183, 34359738367, 9817068105

Offset: 1

Joerg Arndt, Jun 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,9,0,0,-8).

Cf. A213243 (cubes), A213244 (5th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).

[(2^n - 1) / GCD (2^n - 1, 7): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013

A213245:=n->(2^n-1)/gcd(2^n-1,7): seq(A213245(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014

Table[(2^n - 1)/GCD[2^n - 1, 7], {n, 60}] (* Vincenzo Librandi, Mar 16 2013 *)

a(n)=(2^n-1)/gcd(2^n-1,7) \\ Edward Jiang, Sep 04 2014

a(n) = M / gcd( M, 7 ), where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014, verified by Robert Israel, Nov 20 2016: (Start)
a(n) = 9*a(n-3)-8*a(n-6).
G.f.: x*(4*x^4+6*x^3+x^2+3*x+1) / ( (x-1)*(2*x-1)*(x^2+x+1)*(4*x^2+2*x+1) ). (End)

A213246 Number of nonzero elements in GF(2^n) that are 9th powers.

Original entry on oeis.org

1, 1, 7, 5, 31, 7, 127, 85, 511, 341, 2047, 455, 8191, 5461, 32767, 21845, 131071, 29127, 524287, 349525, 2097151, 1398101, 8388607, 1864135, 33554431, 22369621, 134217727, 89478485, 536870911, 119304647, 2147483647, 1431655765, 8589934591, 5726623061, 34359738367, 7635497415

Offset: 1

Joerg Arndt, Jun 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,65,0,0,0,0,0,-64).

Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213247 (11th powers), A213248 (13th powers).

List([1..40],n->(2^n-1)/Gcd(2^n-1,9)); # Muniru A Asiru, Jun 27 2018

[(2^n-1)/GCD(2^n-1, 9): n in [1..40]]; // Vincenzo Librandi, Mar 15 2013

A213246:=n->(2^n-1)/gcd(2^n-1,9): seq(A213246(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014

Table[(2^n - 1)/GCD[2^n - 1, 9], {n, 100}] (* Vincenzo Librandi, Mar 15 2013 *)

a(n)=(2^n-1)/gcd(2^n-1,9) \\ Edward Jiang, Sep 04 2014

a(n) = M / gcd( M, 9 ), where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014: (Start)
a(n) = 65*a(n-6)-64*a(n-12).
G.f.: x*(2*x^2 -x +1)*(16*x^8 +16*x^7 +28*x^6 +16*x^5 +25*x^4 +8*x^3 +7*x^2 +2*x +1) / ((x -1)*(x +1)*(2*x -1)*(2*x +1)*(x^2 -x +1)*(x^2 +x +1)*(4*x^2 -2*x +1)*(4*x^2 +2*x +1)). (End)
Conjectures verified by Robert Israel, Jun 27 2018.

A213247 Number of nonzero elements in GF(2^n) that are 11th powers.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 93, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 95325, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 97612893, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 68719476735

Offset: 1

Joerg Arndt, Jun 07 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1024).

Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213248 (13th powers).

[(2^n - 1) / GCD (2^n - 1, 11): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013

A213247:=n->(2^n-1)/igcd(2^n-1,11): seq(A213247(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014

Table[(2^n - 1)/GCD[2^n - 1, 11], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)

{ for(n=1,36,if(n%10,a=2^n-1,a=(2^n-1)/11);print1(a,", ")) } \\ K. Spage, Aug 23 2014

a(n) = M / GCD( M, 11 ) where M=2^n-1.
From Colin Barker, Aug 24 2014: (Start)
a(n) = 1025*a(n-10)-1024*a(n-20).
G.f.: x*(512*x^18 +768*x^17 +896*x^16 +960*x^15 +992*x^14 +1008*x^13 +1016*x^12 +1020*x^11 +1022*x^10 +93*x^9 +511*x^8 +255*x^7 +127*x^6 +63*x^5 +31*x^4 +15*x^3 +7*x^2 +3*x +1) / (1024*x^20 -1025*x^10 +1).
(End)
a(n) = (2^n - 1)/11 if n is divisible by 10, 2^n - 1 otherwise. - Robert Israel, Aug 24 2014

A213248 Number of nonzero elements in GF(2^n) that are 13th powers.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 315, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 1290555, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 5286113595

Offset: 1

Joerg Arndt, Jun 07 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4097, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4096).

Cf. A213243 (cubes), A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers).

[(2^n - 1) / GCD (2^n - 1, 13): n in [1..40]]; // Vincenzo Librandi, Mar 17 2013

A213248:=n->(2^n-1)/gcd(2^n-1,13): seq(A213248(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014

Table[(2^n - 1)/GCD[2^n - 1, 13], {n, 40}] (* Vincenzo Librandi, Mar 17 2013 *)

a(n)=(2^n-1)/gcd(2^n-1,13) \\ Edward Jiang, Sep 04 2014

a(n) = M / gcd( M, 13 ) where M=2^n-1.
Conjectures from Colin Barker, Aug 24 2014: (Start)
a(n) = 4097*a(n-12)-4096*a(n-24).
G.f.: x*(2048*x^22 +3072*x^21 +3584*x^20 +3840*x^19 +3968*x^18 +4032*x^17 +4064*x^16 +4080*x^15 +4088*x^14 +4092*x^13 +4094*x^12 +315*x^11 +2047*x^10 +1023*x^9 +511*x^8 +255*x^7 +127*x^6 +63*x^5 +31*x^4 +15*x^3 +7*x^2 +3*x +1) / (4096*x^24 -4097*x^12 +1). (End)

A213244 Number of nonzero elements in GF(2^n) that are 5th powers.

Original entry on oeis.org

1, 3, 7, 3, 31, 63, 127, 51, 511, 1023, 2047, 819, 8191, 16383, 32767, 13107, 131071, 262143, 524287, 209715, 2097151, 4194303, 8388607, 3355443, 33554431, 67108863, 134217727

Offset: 1

Joerg Arndt, Jun 07 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 17, 0, 0, 0, -16).

Cf. A213243 (cubes), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).

[(2^n - 1) / GCD (2^n - 1, 5): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013

A213244:=n->(2^n-1)/gcd(2^n-1,5): seq(A213244(n), n=1..40); # Wesley Ivan Hurt, Aug 24 2014

Table[(2^n - 1)/GCD[2^n - 1, 5], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)

a(n)=(2^n-1)/gcd(2^n-1,5) \\ Edward Jiang, Sep 04 2014

a(n) = M / GCD( M, 5 ) where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014: (Start)
a(n) = 17*a(n-4)-16*a(n-8).
G.f.: x*(8*x^6+12*x^5+14*x^4+3*x^3+7*x^2+3*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(x^2+1)*(4*x^2+1)).
(End)

A088840 Denominator of sigma(4n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 127, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1

Offset: 1

Labos Elemer, Nov 04 2003

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

See A088839 for numerator.

Cf. A088837, A088838, A088841, A038712, A080278, A000203, A001620, A007814, A193553, A213243.

Table[Denominator[DivisorSigma[1, 4*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := Module[{e = IntegerExponent[n, 2]}, (((-1)^e+2)*(2^(e+1)-1))/3]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)

A088840(n) = denominator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

a(n) = {my(e = valuation(n, 2)); (((-1)^e+2) * (2^(e+1)-1))/3;} \\ Amiram Eldar, Oct 03 2023

From Amiram Eldar, Oct 03 2023: (Start)
Multiplicative with a(2^e) = (((-1)^e+2)*(2^(e+1)-1))/3 = A213243(e+1), and a(p^e) = 1 for an odd prime p.
a(n) = A213243(A007814(n+1)).
Dirichlet g.f.: ((8^s + 4^s + 2^(s+1))/(8^s + 4^s - 2^(s+2) - 4)) * zeta(s).
Sum_{k=1..n} a(k) = (2*n/(3*log(2))) * (log(n) + gamma - 1 + 7*log(2)/12), where gamma is Euler's constant (A001620). (End)

Typo in definition corrected by Antti Karttunen, Nov 18 2017

A227984 Triangle T(n,k), read by rows: T(n,k) is the numerator of (1-2^(n-k+1))/(1-2^(k+1)).

Original entry on oeis.org

1, 3, 1, 7, 1, 1, 15, 7, 3, 1, 31, 5, 1, 1, 1, 63, 31, 15, 7, 3, 1, 127, 21, 31, 1, 7, 1, 1, 255, 127, 9, 31, 15, 1, 3, 1, 511, 85, 127, 21, 1, 5, 7, 1, 1, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1, 2047, 341, 73, 17, 127, 1, 31, 1, 1, 1, 1, 4095, 2047, 1023

Offset: 0

Vincenzo Librandi, Aug 12 2013

The denominators are given in A228035.

The first column is A000225, the second column is A213243, and the third column is A213245.

			Triangle begins:
1;
3,     1;
7,     1,  1;
15,    7,  3,    1;
31,    5,  1,    1,  1;
63,   31,  15,   7,  3,  1;
127,  21,  31,   1,  7,  1,  1;
255,  127,  9,   31, 15, 1,  3,  1;
511,  85,  127,  21, 1,  5,  7,  1, 1;
1023, 511, 255, 127, 63, 31, 15, 7, 3, 1;
2047, 341, 73,   17, 127, 1, 31, 1, 1, 1, 1; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000225, A213243, A213245, A228035.

[Numerator((1-2^(n-k+1))/(1-2^(k+1))): k in [0..n], n in [0..11]];

a[n_, k_] := Numerator[(1 - 2^(n - k + 1))/(1 - 2^(k + 1))];
Table[a[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

A228035 Triangle T(n,k), read by rows: T(n,k) is the denominator of (1-2^(n-k+1))/(1-2^(k+1)).

Original entry on oeis.org

1, 1, 3, 1, 1, 7, 1, 3, 7, 15, 1, 1, 1, 5, 31, 1, 3, 7, 15, 31, 63, 1, 1, 7, 1, 31, 21, 127, 1, 3, 1, 15, 31, 9, 127, 255, 1, 1, 7, 5, 1, 21, 127, 85, 511, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1, 1, 1, 1, 31, 1, 127, 17, 73, 341, 2047, 1, 3, 7, 15, 31, 63

Offset: 0

Vincenzo Librandi, Aug 12 2013

The numerators are given in A227984.

The first diagonal is A000225, the second diagonal is A213243, the third diagonal is A213245.

			Triangle begins:
1;
1, 3;
1, 1, 7;
1, 3, 7, 15;
1, 1, 1, 5,  31;
1, 3, 7, 15, 31, 63;
1, 1, 7, 1,  31, 21, 127;
1, 3, 1, 15, 31, 9,  127, 255;
1, 1, 7, 5,  1,  21, 127, 85,  511;
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023;
1, 1, 1, 1,  31, 1,  127, 17,  73,  341, 2047; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000225, A213243, A213245, A227984.

[Denominator((1-2^(n-k+1))/(1-2^(k+1))): k in [0..n], n in [0..11]];

a[n_, k_] := Denominator[(1 - 2^(n - k + 1))/(1 - 2^(k + 1))];
Table[a[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

A228147 Triangle T(n,k), read by rows: T(n,k) is the denominator of (1+2^(n-k+1))/(1-2^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 1, 1, 7, 5, 1, 3, 7, 3, 31, 1, 1, 7, 5, 31, 21, 1, 3, 7, 15, 31, 63, 127, 1, 1, 7, 5, 31, 7, 127, 85, 1, 3, 7, 3, 31, 63, 127, 51, 511, 1, 1, 7, 5, 31, 21, 127, 85, 511, 341, 1, 3, 7, 15, 31, 63, 127, 15, 511, 1023, 2047, 1, 1, 7, 5, 31

Offset: 0

Vincenzo Librandi, Aug 15 2013

The numerators are given in A228146.

The first diagonal is A213243, the second diagonal is A213244, the third diagonal is A213246, the fourth diagonal is A213247.

			Triangle begins:
1;
1,1;
1,3,7;
1,1,7,5;
1,3,7,3,31;
1,1,7,5,31,21;
1,3,7,15,31,63,127;
1,1,7,5,31,7,127,85;
1,3,7,3,31,63,127,51,511;
1,1,7,5,31,21,127,85,511,341;
1,3,7,15,31,63,127,15,511,1023,2047;
1,1,7,5,31,21,127,85,511,341,2047,1365; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A213243, A213244, A213245, A213246, A213247, A213248.

[Denominator((1+2^(n-k+1))/(1-2^(k+1))): k in [0..n], n in [0..11]];

a[n_, k_] := Denominator[(1 + 2^(n - k + 1))/(1 - 2^(k + 1))]; Table[a[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

cp's OEIS Frontend

A083420 a(n) = 2*4^n - 1.

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A213245 Number of nonzero elements in GF(2^n) that are 7th powers.

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A213246 Number of nonzero elements in GF(2^n) that are 9th powers.

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A213247 Number of nonzero elements in GF(2^n) that are 11th powers.

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A213248 Number of nonzero elements in GF(2^n) that are 13th powers.

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A213244 Number of nonzero elements in GF(2^n) that are 5th powers.

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A088840 Denominator of sigma(4n)/sigma(n).

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