A395754 Number of k < m^2 such that both rad(k) and rad(k+1) divide m, where rad = A007947 and m = A005117(n).

0, 1, 0, 0, 4, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 10, 0, 0, 2, 0, 0, 1, 0, 0, 9, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 7, 0, 0, 4, 0, 0, 1, 0, 6, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 7, 0, 0, 1, 0, 0, 3, 0, 0, 6, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 1, 0, 5, 0, 0

Offset

1,5

Comments

Number of k < A062503(n) such that both A007947(k) and A007947(k+1) divide A005117(n).

Conjecture: for odd A005117(n), a(n) = 0.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

For A005117(n) = prime p (i.e., for n in A071403), a(n) = 0 except for a(2) = 1.

For even A005117(n) = 2*k (i.e., n in A383008), a(n) >= 1, since rad(1) | 2*k and rad(2) | 2*k.

a(A365435(i)) = A002071(i).

Example

Let s = A005117.
Table of n, s(n), a(n), a(n)+1 for select n:
     n    s(n)                    a(n)  List of k.
  -----------------------------------------------------------------
     1      1                       0   {}
     2      2                       1   {1}
     5      6 = 2 * 3               4   {1, 2, 3, 8}
     7     10 = 2 * 5               2   {1, 4}
    19     30 = 2 * 3 * 5          10   {1, 2, 3, 4, 5, 8, 9, 15, 24, 80}
    28     42 = 2 * 3 * 7           9   {1, 2, 3, 6, 7, 8, 27, 48, 63}
    41     66 = 2 * 3 * 11          7   {1, 2, 3, 8, 11, 32, 242}
    49     78 = 2 * 3 * 13          6   {1, 2, 3, 8, 12, 26}
    69    110 = 2 * 5 * 11          3   {1, 4, 10}
    85    138 = 2 * 3 * 23          5   {1, 2, 3, 8, 23}
   521    858 = 2 * 3 * 11 * 13    11   {1, 2, 3, 8, 11, 12, 26, 32, 143, 242, 351}
   529    870 = 2 * 3 *  5 * 29    12   {1, 2, 3, 4, 5, 8, 9, 15, 24, 29, 80, 144}
  1163   1914 = 2 * 3 * 11 * 29     8   {1, 2, 3, 8, 11, 32, 87, 242}

Mathematica

s = Select[Range[150], SquareFreeQ]; rad[x_] := Times @@ FactorInteger[x][[All, 1]]; f[x_] := Select[Range[x^2, 1, -1], Divisible[x, rad[#]] &]; Table[Count[Partition[f[s[[n]]], 2, 1], _?(#1 - #2 == 1 & @@ # &)], {n, Length[s]}]

Crossrefs

Cf. A002071, A005117, A007947, A039956, A062503, A071403, A365435, A383008, A395721.

Sequence in context: A355997 A136452 A247703 * A280639 A350708 A319037

Adjacent sequences: A395751 A395752 A395753 * A395755 A395756 A395757

Keyword

nonn

Author

Michael De Vlieger, May 05 2026

Status

approved