A063274 - OEIS

%I #14 May 13 2013 01:54:05

%S 1,2,3,1,2,3,4,1,1,2,3,2,2,3,4,1,2,2,3,2,3,3,4,2,1,2,1,2,2,3,2,1,2,2,

%T 2,1,2,3,3,2,2,3,2,2,2,3,3,2,1,2,3,2,2,2,3,3,2,2,2,3,2,3,2,1,2,3,3,2,

%U 3,3,3,1,2,2,3,2,3,3,3,2,1,2,3,3,2,3,4,2,2,2,2,3,3,3,3,2,2,2,2,1,2,3,3,2,3

%N Number of powerful numbers (definition 1) required to sum to n.

%C Heath-Brown proves that a(n) <= 3 for all large n. It seems that n > 119 suffices. - _Charles R Greathouse IV_, Nov 19 2012

%D D. R. Heath-Brown, Ternary quadratic forms and sums of three square-full numbers, Séminaire de Théorie des Nombres, Paris 1986-87, pp. 137-163; Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988.

%H Charles R Greathouse IV, <a href="/A063274/b063274.txt">Table of n, a(n) for n = 1..10000</a>

%e The powerful numbers (A001694) start 1,4,8,9,... 11=1+1+9 and is not the sum of fewer terms, so a(11)=3.

%o (PARI) W=vector(99); W[1]=1; for(n=2,#W, if(ispowerful(n), W[n]=1; next); b=n; for(i=1,n\2, b=min(b,W[i]+W[n-i])); W[n]=b); W \\ _Charles R Greathouse IV_, Nov 19 2012

%Y Cf. A001694, A056828, A063275.

%K nonn

%O 1,2

%A _Jud McCranie_, Jul 13 2001