A092440 -id:A092440 - cp's OEIS frontend

A281482 a(n) = 2^(n + 1) * (2^n + 1) - 1.

3, 11, 39, 143, 543, 2111, 8319, 33023, 131583, 525311, 2099199, 8392703, 33562623, 134234111, 536903679, 2147549183, 8590065663, 34360000511, 137439477759, 549756862463, 2199025352703, 8796097216511, 35184380477439, 140737505132543, 562949986975743

Offset: 0

Jaroslav Krizek, Jan 22 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281481 (2^(n - 1) * (2^n + 1) + 1).

[2^(n + 1) * (2^n + 1) - 1: n in [0..200]];

Vec((3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A006598 Numbers n such that 2^(2n+1) - 2^(n+1) + 1 is a prime.

Original entry on oeis.org

1, 3, 23, 36, 39, 56, 75, 83, 119, 120, 176, 183, 228, 683, 1520

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

These numbers satisfy A100014(n)=2. - Michel Marcus, Mar 07 2013

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A007670.

Indices of primes in A092440. For the actual primes see A325914.

Do[ If[ PrimeQ[ 2^(2n + 1) - 2^(n + 1) + 1 ], Print[n] ], {n, 1, 4000} ]

A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Phi_n(x) is the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nL(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
Let L(n) = the Aurifeuillian L-part of 2^n+1, L(n) = 2^(n/2) - 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let L*(n) = GCD(L(n), J*(n)).
This sequence lists all n such that L*(n) is prime.

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 and is not prime.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 23, 111, 479, 1983, 8063, 32511, 130559, 523263, 2095103, 8384511, 33546239, 134201343, 536838143, 2147418111, 8589803519, 34359476223, 137438429183, 549754765311, 2199021158399, 8796088827903, 35184363700223, 140737471578111, 562949919866879

Offset: 0

Robert Price, Jan 20 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267814.

Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - Jaroslav Krizek, Jan 22 2017

rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 22 2016 and Apr 16 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - Jaroslav Krizek, Jan 22 2017

A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

Original entry on oeis.org

2, 4, 11, 37, 137, 529, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529

Offset: 0

Jaroslav Krizek, Jan 22 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281482 (2^(n + 1) * (2^n + 1) - 1).

Cf. A278930.

[2^(n - 1) * (2^n + 1) + 1: n in [0..200]];

Vec((2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
a(n) = A278930(n - 2) for n >= 7. - Georg Fischer, Mar 26 2019

A220980 a(n) = 5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1: the right Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

71, 19151, 10165751, 6152578751, 3820806643751, 2384948876968751, 1490211490478593751, 931334495635986718751, 582078099253082277343751, 363798066973743438730468751, 227373698726297855377246093751, 142108550062403118610382324218751

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220979.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (781,-101530,2538250,-12203125,9765625).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1, {n, 0, 20}]

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * A220979(n) * a(n).

G.f.: -(27734375*x^4-22687500*x^3+2417450*x^2-36300*x+71) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). [Colin Barker, Jan 03 2013]

A220981 a(n) = 6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1: the left Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

13, 39493, 58809673, 78002205553, 101481622729633, 131604778271166913, 170578072060319947393, 221073129991920857571073, 286511629376393032228157953, 371319255900007820952456748033, 481229795439713382306649129101313

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220982.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (1555,-345210,12427560,-72550080,60466176).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1, {n, 0, 20}]
LinearRecurrence[{1555,-345210,12427560,-72550080,60466176},{13,39493,58809673,78002205553,101481622729633},20] (* Harvey P. Dale, Oct 01 2021 *)

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * a(n) * A220982(n).

G.f.: -(21835008*x^4+24984288*x^3+1885788*x^2+19278*x+13) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

97, 55117, 62169337, 78727802257, 101638351073377, 131638631590149697, 170585384377200633217, 221074709452366968135937, 286511970539849391404729857, 371319329591314394530363646977, 481229811357035602199451623479297

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220981.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (1555,-345210,12427560,-72550080,60466176).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1, {n, 0, 20}]

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * A220981(n) * a(n).

G.f.: -(162922752*x^4-124050528*x^3+9947772*x^2-95718*x+97) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220987 The left Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

58367, 3812903020530517, 107454987376543082369146967, 2808133028073215608147547774721982717, 72885505321551844061773948114862247606146502767, 1890579685660625069233746109183146734516524279847333062117

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220988.

Wikipedia, Cunningham Project

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[z = 11^n; 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1, {n, 0, 10}]

a(n) = 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1 with z = 11^n.

Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * a(n) * A220988(n).

A220988 The right Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

407353, 4572972882642803, 109245858982819139102535553, 2812355783638980226466572392952970603, 72895462357781065526518523423275265184080402953, 1890603163831201090586603020695655490130990020251181357603

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220987.

Wikipedia, Cunningham Project

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[z = 11^n; 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1, {n, 0, 10}]

a(n) = 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1 with z = 11^n.

Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * A220987(n) * a(n).

cp's OEIS Frontend

A281482 a(n) = 2^(n + 1) * (2^n + 1) - 1.

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A006598 Numbers n such that 2^(2n+1) - 2^(n+1) + 1 is a prime.

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A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

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A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

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A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

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A220980 a(n) = 5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1: the right Aurifeuillian factor of 5^(10n+5) - 1.

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A220981 a(n) = 6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1: the left Aurifeuillian factor of 6^(12n+6) + 1.

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A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

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