A213246 -id:A213246 - 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

A213243 Number of nonzero elements in GF(2^n) that are cubes.

Original entry on oeis.org

1, 1, 7, 5, 31, 21, 127, 85, 511, 341, 2047, 1365, 8191, 5461, 32767, 21845, 131071, 87381, 524287, 349525, 2097151, 1398101, 8388607, 5592405, 33554431, 22369621, 134217727, 89478485, 536870911, 357913941, 2147483647, 1431655765, 8589934591, 5726623061, 34359738367, 22906492245

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 (0,5,0,-4).

Cf. A213244 (5th powers), A213245 (7th powers), A213246 (9th powers), A213247 (11th powers), A213248 (13th powers).

[(2^n - 1) / GCD (2^n - 1, 3): n in [1..40]]; // Vincenzo Librandi, Mar 16 2013

A213243:=n->(2^n-1)/gcd(2^n-1,3): seq(A213243(n), n=1..50); # Wesley Ivan Hurt, Aug 23 2014

Table[(2^n - 1)/GCD[2^n - 1, 3], {n, 50}] (* Vincenzo Librandi, Mar 16 2013 *)
LinearRecurrence[{0,5,0,-4},{1,1,7,5},40] (* Harvey P. Dale, Jan 05 2017 *)

a(n)=(2^n-1)/gcd(2^n-1,3) \\ Edward Jiang, Sep 04 2014

a(n) = M / gcd( M, 3 ), where M=2^n-1.
Conjectures from Colin Barker, Aug 23 2014, verified by Robert Israel, Apr 22 2016: (Start)
a(n) = (-1)*((-2+(-1)^n)*(-1+2^n))/3.
a(n) = 5*a(n-2) - 4*a(n-4).
G.f.: x*(2*x^2+x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)). (End)
E.g.f.: (-1 + exp(x) - 2*exp(3*x) + 2*exp(4*x))*exp(-2*x)/3. - Ilya Gutkovskiy, Apr 22 2016

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)

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)

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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A213243 Number of nonzero elements in GF(2^n) that are cubes.

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A213245 Number of nonzero elements in GF(2^n) that are 7th 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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A228147 Triangle T(n,k), read by rows: T(n,k) is the denominator of (1+2^(n-k+1))/(1-2^(k+1)).

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