cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A216835 Fibonacci + Goldbach (dual sequence to A216275). a(1)=5, a(2)=7 and for n>=3, a(n) = g(a(n-1) + a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

Original entry on oeis.org

5, 7, 7, 11, 13, 19, 29, 43, 67, 107, 167, 271, 433, 701, 1129, 1823, 2939, 4759, 7691, 12437, 20123, 32537, 52631, 85121, 137723, 222841, 360551, 583351, 943871, 1527203, 2471071, 3998263, 6469303, 10467547, 16936753, 27404297, 44341027, 71745313, 116086303
Offset: 1

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Author

Vladimir Shevelev, Mar 16 2013

Keywords

Comments

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

Crossrefs

Programs

  • Mathematica
    a[1] = 5; a[2] = 7; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1] + a[n-2]]; Table[a[n], {n,1,100}]

Formula

a(n) = g(A216275(n+2)).

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

Views

Author

Yasutoshi Kohmoto, Dec 28 2002

Keywords

Comments

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)

Crossrefs

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

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

Views

Author

Vladimir Shevelev, Apr 05 2013

Keywords

Crossrefs

Programs

  • Mathematica
    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 *)
Showing 1-3 of 3 results.