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.

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

Offset: 1

Author

M. F. Hasler, Mar 05 2018

Keywords

• nonn
• more

Comments

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.

Examples

			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
		

Crossrefs

Cf. A300004, A292794, A000045.

Programs

PARI

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))

Formula

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

Extensions

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