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%I #38 Mar 18 2023 08:49:14
%S 2,3,4,6,24,114,174,444
%N Numbers k such that exactly one multiset of k positive integers has equal sum and product.
%C No other terms below 10^10 (Ecker, 2002). Probably finite and complete.
%C For any m, there is the multiset {m, 2, 1^(m-2)} with sum and product 2m.
%C (A) If m-1 is composite (m-1=ab), then {a+1, b+1, 1^(m-2)} is another multiset with sum = product. (_Hugo van der Sanden_)
%C (B) If 2m-1 is composite (2m-1=ab), then {2, (a+1)/2, (b+1)/2, 1^(m-3)} is another such multiset. (_Don Reble_)
%C (C) If m = 30j+12, then {2, 2, 2, 2, 2j+1, 1^(30j+7)} is another such multiset. (_Don Reble_)
%C Conditions (A), (B), (C) eliminate all k's except for 2, 3, 4, 6, 30j+0, and 30j+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 Piotr Miska and Maciej Ulas, <a href="https://arxiv.org/abs/2203.03942">On the Diophantine equation sigma_2(Xn)=sigma_n(Xn)</a>, arXiv:2203.03942 [math.NT], 2022.
%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,more
%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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