A213245 -id:A213245 - cp's OEIS frontend

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

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

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)

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

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

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

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A228035 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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