A210256 Differences of the sum of distinct values of {floor(n/k), k=1,...,n}.

2, 1, 3, 1, 4, 1, 2, 4, 2, 1, 6, 1, 2, 2, 6, 1, 3, 1, 7, 2, 2, 1, 4, 6, 2, 2, 3, 1, 9, 1, 3, 2, 2, 2, 10, 1, 2, 2, 4, 1, 10, 1, 3, 3, 2, 1, 5, 8, 3, 2, 3, 1, 4, 2, 11, 2, 2, 1, 6, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 14, 1, 2, 3, 3, 2, 4, 1, 5, 11, 2, 1, 6, 2, 2

Offset: 1

Author

John W. Layman, Mar 19 2012

Keywords

• nonn

Comments

Differences of A051201.

It appears that a(n)=1 if and only if n>1 and n+1 is a prime. For example, the indices where 1 occurs in {a(n)} are {2,4,6,10,12,16,...}. Adding 1 to each of these gives {3,5,7,11,13,17,...} each of which is a prime.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006218, A051201, A055086.

Programs

Maple

b:= proc(n) option remember; add(i, i={seq(floor(n/k), k=1..n)}) end:
a:= n-> b(n+1)-b(n):
seq(a(n), n=1..150); # Alois P. Heinz, Mar 19 2012

Mathematica

b[n_] := b[n] = Total@ Union@ Table[Floor[n/k], {k, 1, n}];
a[n_] := b[n+1] - b[n];
Array[a, 150] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

Python

from math import isqrt
def A210256(n): return ((m:=isqrt((n+1<<2)+1)+1>>1)*(m-1)>>1)+sum((n+1)//k for k in range(1,(n+1)//m+1))-((r:=isqrt((n<<2)+1)+1>>1)*(r-1)>>1)-sum(n//k for k in range(1,n//r+1)) # Chai Wah Wu, Oct 31 2023