

A033179


Numbers k such that exactly one multiset of k positive integers has equal sum and product.


6




OFFSET

1,1


COMMENTS

No other terms below 10^10 (Ecker, 2002). Probably finite and complete.
For any m, there is the multiset {m, 2, 1^(m2)} with sum and product 2m.
(A) If m1 is composite (m1=ab), then {a+1, b+1, 1^(m2)} is another multiset with sum = product. (Hugo van der Sanden)
(B) If 2m1 is composite (2m1=ab), then {2, (a+1)/2, (b+1)/2, 1^(m3)} is another such multiset. (Don Reble)
(C) If m = 30j+12, then {2, 2, 2, 2, 2j+1, 1^(30j+7)} is another such multiset. (Don Reble)
Conditions (A), (B), (C) eliminate all k's except for 2, 3, 4, 6, 30j+0, and 30j+24.


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 174, p. 54, Ellipses, Paris 2008.
R. K. Guy, 'Unsolved Problems in Number Theory' (Section D24).


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CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



