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A033179 Numbers n such that exactly one multiset of n positive integers has equal sum and product. 2


%S 2,3,4,6,24,114,174,444

%N Numbers n such that exactly one multiset of n positive integers has equal sum and product.

%C No other elements below 10^10 (Ecker, 2002). Probably finite and complete.

%C For any n, there is the multiset {n, 2, 1^(n-2)} with sum and product 2n.

%C (A) If n-1 is composite (n-1=ab), then {a+1, b+1, 1^(n-2)} is another multiset with sum = product. (_Hugo van der Sanden_)

%C (B) If 2n-1 is composite (2n-1=ab), then {2, (a+1)/2, (b+1)/2, 1^(n-3)} is another such multiset. (_Don Reble_)

%C (C) If n = 30k+12, then {2, 2, 2, 2, 2k+1, 1^(30k+7)} is another such multiset. (_Don Reble_)

%C Conditions (A), (B), (C) eliminate all n's except for 2, 3, 4, 6, 30k+0, and 30k+24.

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 174, p. 54, Ellipses, Paris 2008.

%D R. K. Guy, 'Unsolved Problems in Number Theory' (Section D24).

%H Michael W. Ecker, <a href="http://www.jstor.org/stable/3219187">When Does a Sum of Positive Integers Equal Their Product?</a> Mathematics Magazine 75(1), 2002, pp. 41-47.

%H Michael A. Nyblom, <a href="https://www.fq.math.ca/Papers1/50-1/Nyblom.pdf">Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem</a>, Fib. Q. 50(1), 2012, 58-61.

%Y Cf. A033178.

%K nonn

%O 1,1

%A _David W. Wilson_

%E Revised by _Don Reble_, Jun 11 2005

%E Edited by _Max Alekseyev_, Nov 13 2013

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