A214156 -id:A214156 - cp's OEIS frontend

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).

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

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; otherwise a(n) is the largest prime divisor of 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, 7, 37, 44, 81, 125, 103, 228, 331, 43, 374, 139, 513, 652, 233, 885, 1118, 2003, 3121, 5124, 8245, 461, 8706, 103, 383, 486, 79, 113, 192, 61, 23, 84, 107, 191, 149, 340, 163, 503, 666, 167, 833, 1000, 1833, 2833, 2333, 5166, 7499

Offset: 0

Vladimir Shevelev, Feb 16 2013

nonn

An analog of the Fibonacci numbers A000045 without semiprimes.

Is the sequence unbounded? (Cf. a dual sequence A214156 which is bounded.)

Peter J. C. Moses, Table of n, a(n) for n = 0..499

Cf. A001358, A214156.

A214094 := proc(n)
    option remember ;
    if n <=1 then
        n;
    else
        a := procname(n-1)+procname(n-2) ;
        if numtheory[bigomega](a) =2 then
            max(op(numtheory[factorset](a)));
        else
            return a;
        end if;
    end if;
end proc: # R. J. Mathar, Feb 18 2013

A214094[0]:=0;
A214094[1]:=1;
A214094[n_]:=A214094[n]=If[PrimeOmega[#]==2,Last[Most[Divisors[#]]],#]&[A214094[n-1]+A214094[n-2]];
Table[A214094[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]] (* Harvey P. Dale, Nov 13 2017 *)

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 *)

A221218 Let sequence B_n={b_m} be defined by: b_1=prime(n), b_2=prime(n+1); for m>=3, b_m=b_(m-2)+b_(m-1) if b_(m-2)+b_(m-1) is not semiprime, otherwise b_m is the least prime divisor of b_(m-2)+b_(m-1). Then a(n) is the maximal term of sequence B_n, or a(n)=0 if B_n is unbounded.

Original entry on oeis.org

570, 570, 570, 570, 19726, 113750, 570, 22534, 570, 570, 570, 570, 399610, 570, 570, 570, 3138, 670, 570, 570, 772, 570, 570, 2448, 109472, 570, 570, 570, 1150, 609, 18644, 71049, 2276, 570, 1634, 1552, 13844, 798, 68830, 6940, 575, 1498, 668, 2551, 1586, 29729, 1748, 113750, 19726, 1435, 194650, 64360, 3213, 953988, 9146, 16539, 811, 8370238, 516878, 881, 99942, 7399, 4160, 215843, 8397, 676, 13397, 1715, 915722, 702, 3572, 141759, 1192, 1131, 762, 24895, 1194, 22534, 1750, 7069, 68830

Offset: 1

Vladimir Shevelev, Feb 22 2013

nonn

Conjecture: All a(n)>=570. Conjecture: All sequences B_n are eventually periodic.
Moreover, our first observations show that up to n=8, the lengths of the periods is 36.
Peter J. C. Moses extended these observations and confirmed the same length 36 of all periods up to n=209.

			In case n=1, B_1 essentially coincides with A214156 and thus a(1)=570 which is the maximal term of A214156.

Cf. A214156.

Terms beginning with a(5) from Peter J. C. Moses

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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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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; otherwise a(n) is the largest prime divisor of 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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A221218 Let sequence B_n={b_m} be defined by: b_1=prime(n), b_2=prime(n+1); for m>=3, b_m=b_(m-2)+b_(m-1) if b_(m-2)+b_(m-1) is not semiprime, otherwise b_m is the least prime divisor of b_(m-2)+b_(m-1). Then a(n) is the maximal term of sequence B_n, or a(n)=0 if B_n is unbounded.

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