A292794 -id:A292794 - cp's OEIS frontend

A300004 First differences of the list A292794 of numbers not congruent to F(k) modulo F(k+1) for any k > 1, where F are the Fibonacci numbers A000045.

4, 2, 4, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 10, 2, 6, 4, 2, 4, 2, 4, 8, 6, 4, 2, 4, 2, 4, 6, 8, 4, 2, 6, 4, 6, 2, 6, 4, 2, 6, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 8, 4, 2, 4, 2, 4, 8, 6, 4, 2, 4, 2, 10, 2, 6, 6, 10, 6, 2, 6, 4, 2, 4, 2, 4, 8, 6, 6, 4, 2, 4, 6, 2, 12, 4, 2, 4

Offset: 0

M. F. Hasler, Feb 25 2018

nonn

The sequence has been chosen 0-indexed, as for A292794.
It has been conjectured that every even number occurs in this sequence.
It would be interesting to know whether there are patterns or formulae for the indices at which the respective even numbers occur.
The index of the first occurrence of 2n in this sequence is listed in A300004 = (1, 0, 5, 26, 14, ...): a(1), a(0), a(5), a(26), a(14), ... is the first occurrence of 2, 4, 6, 8, 10, ... in this sequence.

Cf. A292794, A000045.

Differences@ Join[{0}, Select[Range[3, 500], Function[n, NoneTrue[Block[{k = {1, 1}}, While[Last@k <= n, AppendTo[k, Total@ Take[k, -2]]]; Partition[Most@ k, 2, 1]], Mod[n, #2] == #1 & @@ # &]]]] (* Michael De Vlieger, Mar 19 2018 *)

(D(v)=v[^1]-v[^-1])(A292794=select(is_A292794,[0..500]))

print1(L=0); for(n=1,oo, is_A292794(n)&&print1(",",-L+L=n))

a(n) = A292794(n+1) - A292794(n).

Edited by M. F. Hasler, Mar 05 2018

A300005 Index of first occurrence of 2n in A300004 (or -1 of 2n does not occur), where A300004 are first differences of numbers not congruent to A000045(k) modulo A000045(k+1) for any k > 1.

Original entry on oeis.org

1, 0, 5, 26, 14, 100, 323, 1671, 4293, 10934, 208741, 753123, 627460, 87918559, 1137656208

Offset: 1

M. F. Hasler, Mar 05 2018

It is only conjectured that every even number occurs in sequence A300004.
It would be interesting to know whether there are patterns or formulae for the indices at which the respective even numbers occur in.
As can be seen from a(2) = 0, sequence A300004 has been chosen to be 0-indexed, as for A292794.

			Index n| gap 2n |   a(n)   | A292794(a(n))   with a(n) = the smallest k
   1   |    2   |     1    |       4         such that  A300004(k) = 2n
   2   |    4   |     0    |       0         = A292794(k+1) - A292794(k)
   3   |    6   |     5    |      16
   4   |    8   |    26    |     106
   5   |   10   |    14    |      54
   6   |   12   |    100   |     444
   7   |   14   |    323   |     1456
   8   |   16   |   1671   |     7614
   9   |   18   |   4293   |    19602
  10   |   20   |   10934  |    49966
  11   |   22   |  208741  |    954384
  12   |   24   |  753123  |   3443356
  13   |   26   |  627460  |   2868820
  14   |   28   | 87918559 |  401976096
  15   |   30   |1137656208| 5201526136

Cf. A300004, A292794, A000045.

A300005=List();b=c=L=0;ng=2;for(n=1,oo, is_A292794(n)||next; c++; bittest(b,g=-L+L=n)&&next; b+=2^g; listput(A300005,[g,c-1,n-g]); g>ng&&next; listsort(A300005); for(i=ng/2,#A300005, A300005[i][1]>ng&&break; printf("%d,",A300005[i]);ng+=2))

a(n) = min { k | 2n = A300004(k) = A292794(k+1) - A292794(k) }.

a(14)-a(15) from Jacques Tramu, Feb 26 2018

cp's OEIS Frontend

A300004 First differences of the list A292794 of numbers not congruent to F(k) modulo F(k+1) for any k > 1, where F are the Fibonacci numbers A000045.

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A300005 Index of first occurrence of 2n in A300004 (or -1 of 2n does not occur), where A300004 are first differences of numbers not congruent to A000045(k) modulo A000045(k+1) for any k > 1.

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