A395896 Number of k < m^2 such that both rad(k) and rad(k+1) divide m, where rad = A007947 and m = A019565(n).

0, 1, 0, 4, 0, 2, 0, 10, 0, 2, 0, 9, 0, 4, 0, 23, 0, 1, 0, 7, 0, 3, 0, 18, 0, 2, 0, 14, 0, 7, 0, 40, 0, 1, 0, 6, 0, 4, 0, 18, 0, 3, 0, 14, 0, 7, 0, 39, 0, 1, 0, 11, 0, 6, 0, 31, 0, 3, 0, 24, 0, 12, 0, 68, 0, 2, 0, 7, 0, 3, 0, 16, 0, 3, 0, 13, 0, 6, 0, 36, 0, 2

Offset

0,4

Comments

Number of k < A019565(n)^2 such that both A007947(k) and A007947(k+1) divide A019565(n).

Conjecture: a(n) = 0 for even n (that is, for odd numbers in A019565).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2048

Michael De Vlieger, Log log scatterplot of a((n+1)/2), n = 1..2^11 (eliminating zeros), with axis x = (n+1)/2.

Michael De Vlieger, Fan style binary tree, n = 0..2^11-1, using a color function where gray = 0, blue = 1, ..., magenta = 487.

Formula

a(2^i-1) = A002071(i).

Example

Let r(n) be the reversal of the binary expansion of n (replacing 0 with "." for clarity).
Let s(n) = A019565(n) and b(n) is the largest term in the list of k < A019565(n)^2 such that both rad(k) and rad(k+1) divide A019565(n).
Table of n, r(n), s(n), a(n), b(n) for select n:
   n   r(n)     s(n)   b(n)  a(n)  List of k.
  ----------------------------------------------------------------
   0   .    ->    1      -     -   {}
   1   1    ->    2      1     1   {1}
   2   .1   ->    3      -     -   {}
   3   11   ->    6      8     4   {1, 2, 3, 8}
   4   ..1  ->    5      -     -   {}
   5   1.1  ->   10      4     2   {1, 4}
   6   .11  ->   15      -     -   {}
   7   111  ->   30     80    10   {1, 2, 3, ..., 15, 24, 80}
   9   1..1 ->   14      7     2   {1, 7}
  11   11.1 ->   42     63     9   {1, 2, 3, ..., 27, 48, 63}
  13   1.11 ->   70     49     4   {1, 4, 7, 49}
  15   1111 ->  210   4374    23   {1, 2, 3, ..., 224, 2400, 4374}
  17   1... ->   22      1     1   {1}
  19   11.. ->   66    242     7   {1, 2, 3, 8, 11, 32, 242}
  21   1.1. ->  110     10     3   {1, 4, 10}
  23   111. ->  330    242    18   {1, 2, 3, ..., 99, 120, 242}
  25   1..1 ->  154      7     2   {1, 7}
  27   11.1 ->  462    242    14   {1, 2, 3, ..., 63, 98, 242}
  29   1.11 ->  770    175     7   {1, 4, 7, 10, 49, 55, 175}
  31   1111 -> 2310   9800    40   {1, 2, 3, ..., 3024, 4374, 9800}

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; f[x_] := Select[Range[x^2, 1, -1], Divisible[x, rad[#] ] &]; Monitor[Table[k = Times @@ MapIndexed[Prime[First[#2] ]^#1 &, Reverse@ IntegerDigits[n, 2] ]; Count[Partition[f[k], 2, 1], _?(#1 - #2 == 1 & @@ # &)], {n, 0, 31}], n]

Crossrefs

Cf. A002071, A005117, A019565, A395754.

Sequence in context: A272198 A272203 A324815 * A019200 A324820 A346085

Adjacent sequences: A395893 A395894 A395895 * A395897 A395898 A395899

Keyword

nonn

Author

Michael De Vlieger, May 12 2026

Status

approved