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A033179 Numbers n such that exactly one multiset of n positive integers has equal sum and product. 1
2, 3, 4, 6, 24, 114, 174, 444 (list; graph; refs; listen; history; text; internal format)



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

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

(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)

(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) If n = 30k+12, then {2, 2, 2, 2, 2k+1, 1^(30k+7)} is another such multiset. (Don Reble)

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


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).


Table of n, a(n) for n=1..8.

Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine 75(1), 2002, pp. 41-47.


Cf. A033178.

Sequence in context: A217442 A065199 A249156 * A067244 A084811 A051856

Adjacent sequences:  A033176 A033177 A033178 * A033180 A033181 A033182




David W. Wilson


Revised by Don Reble, Jun 11 2005

Edited by Max Alekseyev, Nov 13 2013



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Last modified November 29 14:32 EST 2015. Contains 264654 sequences.