A175181 -id:A175181 - cp's OEIS frontend

A175289 Pisano period of A002605 modulo n.

1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, 48, 10, 22, 3, 120, 12, 27, 48, 840, 24, 320, 1, 30, 144, 48, 9, 36, 180, 12, 24, 280, 48, 308, 10, 72, 22, 46, 3, 336, 120, 144, 12, 936, 27, 120, 48, 180, 840, 29, 24, 60, 320, 144, 1, 24, 30, 1122, 144

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

a(79)=6240. [John W. Layman, Aug 10 2010]

			Reading 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448,.. modulo 12 gives 0, 1, 2, 6, 4, 8, 0, 4, 8, 0, 4, 8 ,.. with period length a(n=12)= 3.

Michael De Vlieger, Table of n, a(n) for n = 1..200

Cf. A001175, A046738, A175181, A175182, A175183, A175184, A175185, A175286.

a={1};For[n=2,n<=80,n++,{x={{0,1}}; t={1,1}; While[ !MemberQ[x,t], {xl = x[[ -1]]; AppendTo[x,t]; t={Mod[2*(t[[1]]+xl[[1]]),n], Mod[2*(t[[2]] + xl[[2]]),n]};}]; p = Flatten[Position[x,t]][[1]]; AppendTo[a, Length[x] - p+1];}]; Print[a]; (* John W. Layman, Aug 10 2010 *)

Terms beyond a(28)=48 from John W. Layman, Aug 10 2010

A174191 Expansion of (1+x)(2x-1)/((1-x)(x^2+2x-1)).

Original entry on oeis.org

1, 2, 3, 6, 13, 30, 71, 170, 409, 986, 2379, 5742, 13861, 33462, 80783, 195026, 470833, 1136690, 2744211, 6625110, 15994429, 38613966, 93222359, 225058682, 543339721, 1311738122, 3166815963, 7645370046, 18457556053, 44560482150, 107578520351

Offset: 0

Clark Kimberling, Mar 11 2010

nonn

Pisano period lengths: 1, 2, 8, 4, 12, 8, 6, 8, 24, 12, 24, 8, 28, 6, 24, 16, 16, 24, 40, 12,.. (is this A175181?) - R. J. Mathar, Aug 10 2012

			a(2) = 2*a(1) + a(0) - 2 = 2*2 + 1 - 2 = 3
a(3) = 2*a(2) + a(1) - 2 = 2*3 + 2 - 2 = 6.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A174192, A001333 (first differences).

LinearRecurrence[{3, -1, -1}, {1, 2, 3}, 31] (* Robert P. P. McKone, Apr 03 2022 *)

a(n) = 2*a(n-1) + a(n-2) - 2, with a(0)=1, a(1)=2.
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = A052937(n-1), n > 0.
a(n) = 3*a(n-1) - a(n-2) - a(n-3). (End)

A175290 Pisano period of A030195 modulo n.

Original entry on oeis.org

1, 3, 1, 3, 4, 3, 42, 6, 1, 12, 120, 3, 84, 42, 4, 12, 16, 3, 90, 12, 42, 120, 176, 6, 20, 84, 1, 42, 280, 12, 480, 24, 120, 48, 84, 3, 36, 90, 84, 12, 40, 42, 42, 120, 4, 528, 46, 12, 294, 60, 16, 84, 2808, 3, 120, 42, 90, 840, 58, 12, 310, 480, 42, 48, 84, 120, 33, 48, 176, 84, 5040, 6, 888, 36, 20, 90, 840, 84, 39, 12

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

			Reading 0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893 .. modulo n=3 gives 0, 1, 0, 0, 0, 0, 0, .. with period length a(n=3)= 1.
Reading modulo n=6 gives 0, 1, 3, 0, 3, 3, 0, 3, 3 with period length a(n=6)=3.

Cf. A001175, A046738, A175181 - A175185, A175286.

A175291 Pisano period of A006130 modulo n.

Original entry on oeis.org

1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, 24, 120, 22, 6, 120, 156, 9, 24, 28, 24, 240, 24, 120, 48, 24, 6, 171, 90, 156, 24, 336, 24, 42, 120, 24, 66, 736, 12, 168, 120, 16, 156, 52, 9, 120, 24, 90, 84, 3480, 24, 20, 240, 24, 48, 312, 120, 748, 48, 22, 24, 5040, 6, 888, 171, 120, 90, 120, 156, 39

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

			Reading 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, ... modulo n=7 gives 0, 1, 1, 4, 0, 5, 5, 6, 0, 4, 4, 2, 0, 6, 6, 3, 0, 2, 2, 1, 0, 3, 3, 5, 0, 1, 1, 4, 0, 5, 5, 6, 0, 4, 4, 2, 0, 6, 6, 3, 0, ... with period a(n=7)=24.

Cf. A001175, A046738, A175181, A175182, A175183, A175184, A175185, A175286.

a(9) corrected by R. J. Mathar, Apr 18 2010

A247250 Indices of Pell numbers having exactly one primitive prime factor.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 24, 29, 30, 32, 33, 35, 38, 41, 42, 50, 53, 54, 56, 58, 59, 66, 69, 89, 90, 94, 95, 97, 99, 101, 104, 117, 118, 120, 135, 138, 160, 167, 181, 191, 210, 221, 237, 242, 247

Offset: 1

Eric Chen, Nov 29 2014

nonn

Conjecture: The n-th Pell number A000129(n) has a primitive prime factor for all n > 1. (The n-th Fibonacci number A000045(n) has a primitive prime factor for all n except n = 0, 1, 2, 6, and 12.)

For prime p, all prime factors of Pell(p) are primitive. Hence the only primes in this sequence are the prime numbers in A096650, which gives the indices of prime Pell numbers.

			Pell(1) = 1, which has no prime factors, so 1 is not in this sequence.
Pell(4) = 12 = 2^2 * 3, but 2 is not a primitive prime factor, and 3 is the only primitive prime factor of Pell(4), so 4 is in this sequence.
Pell(5) = 29, which is a prime and the only primitive prime factor of itself, so 5 is in this sequence.
Pell(12) = 13860 = 2^2 * 3^2 * 5 * 7 * 11, but none of 2, 3, 5, 7 is a primitive prime factor, and 11 is the only primitive prime factor of Pell(12), so 12 is in this sequence.
Pell(14) = 80782 = 2 * 13^2 * 239, but neither 2 nor 13 is a primitive prime factor, and 239 is the only primitive prime factor of Pell(14), so 14 is in this sequence.
Pell(19) = 6625109 = 37 * 179057, both of which are primitive prime factors of Pell(19), so 19 is not in this sequence.

Cf. A152012 (for Fibonacci numbers).

Cf. A000129, A246556, A096650, A086383, A008555, A175181.

Select[Range[1000], PrimePowerQ[(1-Sqrt[2])^EulerPhi[#]*Cyclotomic[#, (1+Sqrt[2])/(1-Sqrt[2])]/GCD[Cyclotomic[#, (1+Sqrt[2])/(1-Sqrt[2])], # ]]&] - Eric Chen, Dec 12 2014
pell[n_] := pell[n] = ((1+Sqrt[2])^n-(1-Sqrt[2])^n )/(2*Sqrt[2]) // Round; primitivePrimeFactors[n_] := Cases[FactorInteger[pell[n]][[All, 1]], p_ /; And @@ (GCD[p, #] == 1 & /@ Array[pell, n-1])]; Reap[For[n=2, n <= 200, n++, If[Length[primitivePrimeFactors[n]] == 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2014 *)

pell(n) = imag((1 + quadgen(8))^n);
isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
lista(nn) = {vp = []; for (n=2, nn, pf = factor(pell(n))[,1]; if (isok(pf, vp), print1(n, ", ")); vp = vecsort(concat(vp, pf),, 8););} \\ Michel Marcus, Nov 29 2014

Two incorrect terms (72 and 110) deleted by Colin Barker, Nov 29 2014

More terms from Colin Barker, Nov 30 2014

cp's OEIS Frontend

A175289 Pisano period of A002605 modulo n.

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A174191 Expansion of (1+x)*(2*x-1)/((1-x)*(x^2+2*x-1)).

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A175290 Pisano period of A030195 modulo n.

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A175291 Pisano period of A006130 modulo n.

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A247250 Indices of Pell numbers having exactly one primitive prime factor.

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Extensions

A174191 Expansion of (1+x)(2x-1)/((1-x)(x^2+2x-1)).