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A336683 Sum of 2^k for all residues k found in the Fibonacci sequence mod n.

%I A336683 #10 Oct 05 2020 18:00:01
%S A336683 1,3,7,15,31,63,127,175,511,1023,1327,4031,7471,16383,32767,43951,
%T A336683 127807,238895,502063,1048575,1319215,2719023,7798639,10692015,
%U A336683 33554431,61209903,134217727,259173375,337649967,1073741823,1571892655,2880154543,5417565487,15638470959
%N A336683 Sum of 2^k for all residues k found in the Fibonacci sequence mod n.
%C A336683 Row n of A079002 compactified as a binary number.
%H A336683 Michael De Vlieger, <a href="/A336683/b336683.txt">Table of n, a(n) for n = 1..3322</a>
%H A336683 Michael De Vlieger, <a href="/A336683/a336683.png">Plot of bits of a(n)</a> beginning with 2^0 at left for 1 <= n <= 5000.
%F A336683 a(n) = Sum(2^k) for all k in row n of A189768.
%F A336683 a(n) = 2^(n+1) - 1 for n in A079002.
%e A336683 a(1) = 1 by convention.
%e A336683 a(2) = 3 = 2^0 + 2^1, since the Fibonacci sequence mod 2 is {0,1,1} repeated, and 0 and 1 appear in the sequence.
%e A336683 a(8) = 175 = 2^0 + 2^1 + 2^2 + 2^3 + 2^5 + 2^7, since the Fibonacci sequence mod 8 is {0,1,1,2,3,5,0,5,5,2,7,1} repeated, and we are missing 4 and 6, leaving the exponents of 2 as shown.
%e A336683 Binary equivalents of first terms:
%e A336683    n    a(n)   a(n) in binary
%e A336683    --------------------------
%e A336683    1      1                 1
%e A336683    2      3                11
%e A336683    3      7               111
%e A336683    4     15              1111
%e A336683    5     31             11111
%e A336683    6     63            111111
%e A336683    7    127           1111111
%e A336683    8    175          10101111
%e A336683    9    511         111111111
%e A336683   10   1023        1111111111
%e A336683   11   1327       10100101111
%e A336683   12   4031      111110111111
%e A336683   13   7471     1110100101111
%e A336683   14  16383    11111111111111
%e A336683   15  32767   111111111111111
%e A336683   16  43951  1010101110101111
%e A336683   ...
%t A336683 {1}~Join~Array[Block[{w = {0, 1}}, Do[If[SequenceCount[w, {0, 1}] == 1, AppendTo[w, Mod[Total@ w[[-2 ;; -1]], #]], Break[]], {i, 2, Infinity}]; Total[2^Union@ w]] &, 33, 2]
%t A336683 (* Second program: generate the first n terms using the plot in Links *)
%t A336683 With[{n = 34, img = ImageData@ ColorNegate@ Import["https://oeis.org/A336683/a336683.png"]}, Map[FromDigits[#, 2] &@ Drop[#, LengthWhile[#, # == 0 &]] &@ Reverse[IntegerPart[#]] &, img[[1 ;; n]]]] (* _Michael De Vlieger_, Oct 05 2020 *)
%Y A336683 Cf. A000045, A001175, A066853, A079002, A128924, A189768.
%K A336683 nonn,easy
%O A336683 1,2
%A A336683 _Michael De Vlieger_, Oct 04 2020