%I #28 Feb 07 2023 12:00:27
%S 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,
%T 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,
%U 3,3,3,3,1,1,1,1,2,3,2,2
%N Number of ways of representing n as the sum of one or more consecutive squarefree numbers.
%H Robert Israel, <a href="/A242667/b242667.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%p A242667 := proc(n)
%p a := 0 ;
%p for i from 1 do
%p if A005117(i) > n then
%p return a;
%p end if;
%p for k from i do
%p su := add(A005117(s),s=i..k) ;
%p if su = n then
%p a := a+1 ;
%p elif su > n then
%p break;
%p fi ;
%p end do:
%p end do:
%p end proc:
%p seq(A242667(n),n=1..80) ; # _R. J. Mathar_, Jun 12 2014
%p # Alternative:
%p N:= 1000:# to get the first N entries
%p A005117:= select(numtheory:-issqrfree,[$1..N]):
%p M:= nops(A005117);
%p A:= Array(1..N):
%p t0:= 0:
%p for n from 1 to M-1 do
%p t0:= t0 + A005117[n];
%p t:= t0;
%p for i from 1 while t <= N do
%p A[t] := A[t]+1;
%p if n+i > M then break fi;
%p t:= t + A005117[n+i]-A005117[i];
%p od;
%p od:
%p seq(A[i],i=1..N); # _Robert Israel_, Jun 25 2014
%t With[{N = 100}, (* to get the first N entries *)
%t A005117 = Select[Range[N], SquareFreeQ];
%t M = Length[A005117];
%t A = Table[0, {N}];
%t t0 = 0;
%t For[n = 1, n <= M-1, n++,
%t t0 = t0+A005117[[n]];
%t t = t0;
%t For[i = 1, t <= N, i++,
%t A[[t]] = A[[t]]+1;
%t If[n+i > M, Break[]];
%t t = t + A005117[[n+i]] - A005117[[i]]]
%t ]
%t ];
%t A (* _Jean-François Alcover_, Feb 07 2023, after _Robert Israel_ *)
%Y Cf. A005117.
%K nonn
%O 1,3
%A _Irina Gerasimova_, May 20 2014