A188795 a(n) counts all integers k in [2,floor(sqrt(n))] such that the number of divisors d>1 of n-k with k|(n-d) equals A188550(n).

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 2, 2, 3, 2, 1, 1, 1, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 4

Author

Vladimir Shevelev, Apr 10 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..10000

Crossrefs

Cf. A188550, A188579, A188794.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local c, h, k, m;
       m, c:= 0, 0;
       for k from 2 to floor(sqrt(n)) do
          h:= nops(select(x-> irem(x, k)=0,
              [seq (n-d, d=divisors(n-k) minus{1})]));
          if h=m then c:=c+1 elif h>m then m, c:= h, 1 fi
       od; c
    end:
seq(a(n), n=4..120);  # Alois P. Heinz, Apr 10 2011

Mathematica

b[n_] := Max @ Table[Length @ Select[Table[n-d, {d, Divisors[n-k] // Rest} ], Mod[#, k] == 0&], {k, 2, Floor[Sqrt[n]]}];
a[n_] := a[n] = Count[Range[2, Floor[Sqrt[n]]], k_ /; Count[Rest @ Divisors[n-k], d_ /; Divisible[n-d, k]] == b[n]];
Table[a[n], {n, 4, 120}] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)