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A106633
Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].
5
1, 4, 8, 12, 17, 21, 27, 31, 37, 42, 48, 52, 60, 64, 70, 76, 83, 87, 95, 99, 107, 113, 119, 123, 133, 138, 144, 150, 158, 162, 172, 176, 184, 190, 196, 202, 213, 217, 223, 229, 239, 243, 253, 257, 265, 273, 279, 283, 295, 300, 308, 314, 322, 326, 336, 342, 352
OFFSET
0,2
COMMENTS
Number of ordered triples [k,l,m] with n = k+l*m and k, l, m all in the range [0..n].
From R. J. Mathar, Jun 30 2013: (Start)
A010766 is the following array A read by antidiagonals:
1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, ...
3, 2, 1, 1, 1, 1, ...
4, 2, 2, 1, 1, 1, ...
5, 3, 2, 2, 1, 1, ...
6, 3, 2, 2, 2, 1, ...
and apparently a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
LINKS
FORMULA
From Ridouane Oudra, Apr 22 2024: (Start)
a(n) = 2*n + 1 + Sum_{k=1..n} floor(n/k);
a(n) = 2*n + 1 + Sum_{k=1..n} tau(k);
a(n) = A005408(n) + A006218(n). (End)
EXAMPLE
0+1*2 = 0+2*1 = 1+1*1 = 2+0*0 = 2+0*1 = 2+0*2 = 2+1*0 = 2+2*0 = 2, so a(2)=8.
a(3)=12: 3+0*0, 3+0*m (6), 2+1*1, 1+2*1 (2), 0+3*1 (2).
MAPLE
A106633 := proc(n)
local a, k, l, m ;
a := 0 ;
for k from 0 to n do
for l from 0 to n do
if l = 0 then
if k = n then
a := a+n+1 ;
end if;
else
m := (n-k)/l ;
if type(m, 'integer') then
a := a+1 ;
end if;
end if;
end do:
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2012
MATHEMATICA
A106633[n_] := Module[{a, m}, a = 0; Do[If[l == 0, If[k == n, a = a + n + 1], m = (n - k)/l; If[IntegerQ[m], a = a + 1]], {k, 0, n}, {l, 0, n}]; a];
Table[A106633[n], {n, 0, 56}] (* Jean-François Alcover, Jun 10 2023, after R. J. Mathar *)
PROG
(PARI) list(n)={
my(v=vector(n), t);
for(i=2, n, for(j=1, min(n\i, i-1), v[i*j]+=2));
for(i=1, sqrtint(n), v[i^2]++);
concat(1, vector(n, k, 2*k+1+t+=v[k]))
}; \\ Charles R Greathouse IV, Oct 17 2012
(Python)
from math import isqrt
def A106633(n): return 1-(s:=isqrt(n))**2+(n+sum(n//k for k in range(1, s+1))<<1) # Chai Wah Wu, Jan 28 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, May 06 2005
EXTENSIONS
Definition clarified by N. J. A. Sloane, Jul 07 2012
STATUS
approved