A214094 -id:A214094 - cp's OEIS frontend

A214156 Dual to A214094: a(0)=0, a(1)=1; a(n) = a(n-1) + a(n-2) if a(n-1) + a(n-2) is not semiprime; a(n) is minimal prime divisor of a(n-1) + a(n-2) if a(n-1) + a(n-2) is semiprime.

0, 1, 1, 2, 3, 5, 8, 13, 3, 16, 19, 5, 24, 29, 53, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 2, 557, 13, 570, 11, 7, 18, 5, 23, 28, 3, 31, 2, 3, 5, 8, 13, 3, 16, 19, 5, 24, 29, 53, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 2, 557, 13, 570, 11, 7, 18

Offset: 0

Vladimir Shevelev, Feb 16 2013

nonn

The sequence has period of length 36: {2,3,5,...,28,3,31} and thus is bounded.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A214094.

A214156[0]:=0; A214156[1]:=1; A214156[n_] := A214156[n] = If[PrimeOmega[#] == 2, First[Rest[Divisors[#]]], #]& [A214156[n-1] + A214156[n-2]]; Table[A214156[n], {n,0,99}] (* Peter J. C. Moses, Feb 18 2013 *)
nxt[{a_,b_}]:={b,If[PrimeOmega[a+b]==2,FactorInteger[a+b][[1,1]],a+b]}; NestList[nxt,{0,1},70][[All,1]] (* or *) PadRight[{0,1,1},70,{28,3,31,2,3,5,8,13,3,16,19,5,24,29,53,2,5,7,12,19,31,50,81,131,212,343,555,2,557,13,570,11,7,18,5,23}] (* Harvey P. Dale, Feb 02 2017 *)

A078414 a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 3, 16, 19, 5, 24, 29, 53, 82, 135, 31, 166, 197, 363, 80, 443, 523, 138, 661, 799, 1460, 2259, 3719, 122, 3841, 3963, 7804, 1681, 1355, 3036, 4391, 1061, 5452, 6513, 11965, 18478, 4349, 3261, 7610, 1553, 187, 1740, 1927, 3667, 5594, 27

Offset: 1

Yasutoshi Kohmoto, Dec 28 2002

From Vladimir Shevelev, Apr 01 2013; edited by Danny Rorabaugh, Feb 19 2016: (Start)
If we consider Fibonacci-like numbers {F_p(n)} without positive multiples of p, where p is a fixed prime, then {F_2(n)} has period of length 1, {F_3(n)} has period of length 3, {F_5(n)} has period of length 6. This sequence is the first which does not have a trivial period and, probably, even is non-periodic.
An open question: Is this sequence bounded?
Consider Fibonacci-like sequences without multiples of several primes, defined analogously: e.g., for {F_(p,q)(n)}, a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2) divided by the maximal possible powers of p and q.
Problem: For what sets of primes is the corresponding Fibonacci-like sequence without multiples of these primes periodic?
Examples: sequence {F_(7,11,13)(n)} has period of length 12: 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 19, 29, 48, 1, 1, 2, 3, 5,...; {F_(11,13,19)(n)} has period of length 9; {F_(13,19,23)(n)} has period of length 12; {F_(17,19,23,29)(n)} has period of length 15; {F_(19,23,31,53,59,89)(n)} has period of length 24; {F_(23,29,73,233)(n)} has period of length 18.
Don Reble noted that lengths of all such periods could only be multiples of 3 because every Fibonacci-like sequence considered here modulo 2 has the form 0,1,1,0,1,1,... .
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..4000
B. Avila, T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5.

Cf. A000045, A078412, A214094, A214156, A216231, A216275, A216835.

a:= proc(n) option remember; local t, j;
      if n<3 then 1
    else t:= a(n-1)+a(n-2);
         while irem(t, 7, 'j')=0 do t:=j od; t
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 25 2012

nxt[{a_,b_}]:=Module[{n=IntegerExponent[a+b,7]},{b,(a+b)/7^n}]; Transpose[ NestList[nxt,{1,1},60]][[1]] (* Harvey P. Dale, Jul 23 2012 *)

a(n) = A242603(a(n-1)+a(n-2)). - R. J. Mathar, Mar 13 2024

Corrected by Harvey P. Dale, Jul 23 2012

A224382 Fibonacci-like numbers without positive multiples of 4: a(0) = 0, a(1) = 1, for n>=2, a(n) = a(n-1) + a(n-2) divided by maximal possible power of 4.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 7, 9, 1, 10, 11, 21, 2, 23, 25, 3, 7, 10, 17, 27, 11, 38, 49, 87, 34, 121, 155, 69, 14, 83, 97, 45, 142, 187, 329, 129, 458, 587, 1045, 102, 1147, 1249, 599, 462, 1061, 1523, 646, 2169, 2815, 1246, 4061, 5307, 2342, 7649, 9991, 4410

Offset: 0

Vladimir Shevelev, Apr 05 2013

nonn

Peter J. C. Moses, Table of n, a(n) for n = 0..9999
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614, 2014 and J. Int. Seq. 17 (2014) # 14.8.5

Cf. A000045, A078414, A214094, A214156, A216231, A216275, A216835.

a[0]:=0; a[1]:=1; a[n_]:=a[n]=#/4^IntegerExponent[#,4]&[(a[n-1]+a[n-2])]; Map[a,Range[0,99]] (* Peter J. C. Moses, Apr 05 2013 *)

A254032 a(0)=0, a(1)=1; for n > 2, a(n) is the smallest prime factor of a(n-1) + a(n-2) not already in the sequence or, if there is no such prime, a(n) = a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 7, 20, 27, 47, 37, 84, 11, 19, 30, 49, 79, 128, 23, 151, 29, 180, 209, 389, 598, 987, 317, 163, 480, 643, 1123, 883, 17, 900, 131, 1031, 83, 557, 640, 1197, 167, 31, 198, 229, 61, 290, 351, 641, 992, 71, 1063, 1134, 2197, 3331, 691

Offset: 0

David S. Newman, Jan 22 2015

nonn

From Kellen Myers, May 10 2015: (Start)
Empirically this sequence grows slower than A000045(n), the Fibonacci sequence, but faster than log(A000045(n)).
Note that in the case where no suitable prime divisor exists, a(n) must take the value a(n-1) + a(n-2) regardless of whether it appears previously. This allows for repetition, e.g., a(86)=a(99)=957. Among the first 1000 terms, there are 9 values a(n) takes twice. (End)

			The first nonprime Fibonacci number is F(5)=8, and so this is the first place that a(n) could disagree with F(n). However, the only prime factor of 8 is 2, which appears as a(2), and thus a(5) must be 8.
For n=7, a(n-1) + a(n-2) = 21. The prime factors of 21 are 3 and 7, and 7 has not yet appeared, so a(7)=7.

Kellen Myers, Table of n, a(n) for n = 0..999

Cf. A000045, A214094.

a[n_] := a[n] =
   Module[{set = seq[n - 1], val = a[n - 1] + a[n - 2], p},
    p = 2;
    While[(Mod[val, p] != 0 || MemberQ[set, p]) && p <= val,
     p = NextPrime[p]
     ];
    If[p > val, Return[val], Return[p]];
    ];
seq[n_] := seq[n] = Append[seq[n - 1], a[n]]
a[1] = 0; a[2] = 1;
seq[2] = {0, 1};
(* Kellen Myers, May 10 2015 *)

Clarification of definition, examples by Kellen Myers, May 10 2015

cp's OEIS Frontend

A214156 Dual to A214094: a(0)=0, a(1)=1; a(n) = a(n-1) + a(n-2) if a(n-1) + a(n-2) is not semiprime; a(n) is minimal prime divisor of a(n-1) + a(n-2) if a(n-1) + a(n-2) is semiprime.

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A078414 a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2).

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A224382 Fibonacci-like numbers without positive multiples of 4: a(0) = 0, a(1) = 1, for n>=2, a(n) = a(n-1) + a(n-2) divided by maximal possible power of 4.

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A254032 a(0)=0, a(1)=1; for n > 2, a(n) is the smallest prime factor of a(n-1) + a(n-2) not already in the sequence or, if there is no such prime, a(n) = a(n-1) + a(n-2).

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