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

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

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• more
• changed

Comments

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

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

(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)

(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) 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).

Links

• Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine 75(1), 2002, pp. 41-47.
• Hlib Husarov and Eberhard Mayerhofer, An algorithm for the Product-Sum Equality, arXiv:2508.09647 [math.NT], 2025. See p. 2.
• Piotr Miska and Maciej Ulas, On the Diophantine equation sigma_2(Xn)=sigma_n(Xn), arXiv:2203.03942 [math.NT], 2022.
• Michael A. Nyblom, Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem, Fib. Q. 50(1), 2012, 58-61.
• Burkard Polster, What's the next freak identity? A new deep connection with Sophie Germain primes, YouTube Mathologer video, 2024.

Crossrefs

Cf. A033178.

Extensions

Revised by Don Reble, Jun 11 2005

Edited by Max Alekseyev, Nov 13 2013