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

1,3

Links

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

Example

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.

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 *)

Crossrefs

Cf. A005117.

Sequence in context: A353322 A158950 A213013 * A059581 A344319 A236998

Adjacent sequences: A242664 A242665 A242666 * A242668 A242669 A242670

Keyword

nonn

Author

Irina Gerasimova, May 20 2014

Status

approved