A033178 Number of multisets of n positive integers with equal sum and product.

1, 1, 1, 3, 1, 2, 2, 2, 2, 3, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 1, 5, 4, 3, 3, 5, 2, 4, 3, 5, 2, 3, 2, 6, 3, 3, 4, 7, 2, 5, 2, 4, 4, 5, 2, 5, 4, 4, 3, 7, 2, 5, 4, 5, 4, 4, 2, 9, 3, 4, 4, 7, 2, 5, 5, 4, 3, 6, 3, 9, 4, 3, 3, 6, 3, 5, 2, 7, 4, 5, 2, 10, 5, 4, 5, 8, 2, 6, 3, 6, 3, 6, 5, 6, 5, 4, 5, 8, 3, 6, 3, 5

Offset: 2

Author

David W. Wilson

Keywords

• nonn

Comments

The multiset {n^1, 2^1, 1^(n-2)} has n elements and sum = product = 2n. Hence a(n) >= 1.

Examples

			a(5) = 3: {2,2,2,1,1}, {3,3,1,1,1}, {5,2,1,1,1}.
a(7) = 2: {4,3,1,1,1,1,1}, {7,2,1,1,1,1,1}.
		

References

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

Links

• David Radcliffe, Table of n, a(n) for n = 2..10000
• Onno M. Cain, Bioperational Multisets in Various Semi-rings, arXiv:1908.03235 [math.RA], 2019.
• L. Kurlandchik and A. Nowicki, When the sum equals the product, The Mathematical Gazette, 84(499) (2000), 91-94. doi:10.2307/3621488.
• Burkard Polster, What's the next freak identity? A new deep connection with Sophie Germain primes, YouTube Mathologer video, 2024.

Crossrefs

Cf. A033179, A104173.