A063274 Number of powerful numbers (definition 1) required to sum to n.

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, 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, 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

Offset: 1

Author

Jud McCranie, Jul 13 2001

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• 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.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A056828, A063275.

Programs

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