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Differences of the sum of distinct values of {floor(n/k), k=1,...,n}.
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%I #24 Oct 31 2023 12:30:05

%S 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,

%T 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,

%U 3,2,4,1,14,1,2,3,3,2,4,1,5,11,2,1,6,2,2

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

%C Differences of A051201.

%C 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.

%H Alois P. Heinz, <a href="/A210256/b210256.txt">Table of n, a(n) for n = 1..10000</a>

%p b:= proc(n) option remember; add(i, i={seq(floor(n/k), k=1..n)}) end:

%p a:= n-> b(n+1)-b(n):

%p seq(a(n), n=1..150); # _Alois P. Heinz_, Mar 19 2012

%t b[n_] := b[n] = Total@ Union@ Table[Floor[n/k], {k, 1, n}];

%t a[n_] := b[n+1] - b[n];

%t Array[a, 150] (* _Jean-François Alcover_, Nov 20 2020, after _Alois P. Heinz_ *)

%o (Python)

%o from math import isqrt

%o 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

%Y Cf. A006218, A051201, A055086.

%K nonn

%O 1,1

%A _John W. Layman_, Mar 19 2012