OFFSET
1,1
COMMENTS
Mollin and Walsh conjectured that there are no further terms.
Heath-Brown proved that the sequence is finite.
No other terms less than 40000000. - Paul.Jobling(AT)WhiteCross.com, May 14 2001
"Powerful numbers" here means "Definition (1)" (A001694): if a prime p divides n then p^2 must also divide n (also called squareful or square-full). - M. F. Hasler, Jan 08 2026
REFERENCES
D. R. Heath-Brown, "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Séminaire de Théorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhauser, pp. 137-163, 1988.
LINKS
T. F. Bloom, Erdős Problem #1107, personal website ErdosProblems.com, as of Nov. 18, 2025.
R. A. Mollin and P. G. Walsh, On Powerful Numbers, Intern. J. Math. and Math. Sci, 9:801-806, 1986.
Eric Weisstein's World of Mathematics, Powerful Number.
EXAMPLE
Smallest powerful numbers are 1, 4, 8, 9, 16, 25,... so 7, 15 and 23 are not the sum of one, two or three of them.
CROSSREFS
KEYWORD
fini,nonn,changed
AUTHOR
Henry Bottomley, Aug 30 2000
STATUS
approved