A242667 Number of ways of representing n as the sum of one or more consecutive squarefree numbers.

1, 1, 2, 0, 2, 2, 1, 1, 0, 2, 3, 0, 2, 2, 1, 1, 3, 1, 1, 0, 3, 1, 3, 2, 0, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 1, 0, 2, 1, 2, 0, 4, 0, 3, 2, 3, 0, 3, 2, 1, 1, 2, 3, 2, 0, 3, 3, 3, 3, 1, 1, 1, 1, 2, 3, 2, 2

Offset: 1

Author

Irina Gerasimova, May 20 2014

Keywords

• nonn

Examples

			a(6)=2 because n=6 itself is already a squarefree number (sum of 1 term), and 6 can in addition be written as A005117(1)+ A005117(2)+A005117(3), a sum of 3 consecutive squarefree numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117.

Programs

Maple

A242667 := proc(n)
    a := 0 ;
    for i from 1 do
        if A005117(i) > n then
            return a;
        end if;
        for k from i do
            su := add(A005117(s),s=i..k) ;
            if su = n then
                a := a+1 ;
            elif su > n then
                break;
            fi ;
        end do:
    end do:
end proc:
seq(A242667(n),n=1..80) ; # R. J. Mathar, Jun 12 2014
# Alternative:
N:= 1000:# to get the first N entries
A005117:= select(numtheory:-issqrfree,[$1..N]):
M:= nops(A005117);
A:= Array(1..N):
t0:= 0:
for n from 1 to M-1 do
  t0:= t0 + A005117[n];
  t:= t0;
  for i from 1 while t <= N do
     A[t] := A[t]+1;
     if n+i > M then break fi;
     t:= t + A005117[n+i]-A005117[i];
  od;
od:
seq(A[i],i=1..N); # Robert Israel, Jun 25 2014

Mathematica

With[{N = 100}, (* to get the first N entries *)
A005117 = Select[Range[N], SquareFreeQ];
M = Length[A005117];
A = Table[0, {N}];
t0 = 0;
For[n = 1, n <= M-1, n++,
   t0 = t0+A005117[[n]];
   t = t0;
   For[i = 1, t <= N, i++,
      A[[t]] = A[[t]]+1;
      If[n+i > M, Break[]];
      t = t + A005117[[n+i]] - A005117[[i]]]
   ]
];
A (* Jean-François Alcover, Feb 07 2023, after Robert Israel *)