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

2,4

Comments

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

Example

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}.

Crossrefs

Cf. A033179, A104173.

Sequence in context: A305516 A305182 A291047 * A029418 A185736 A144148

Adjacent sequences: A033175 A033176 A033177 * A033179 A033180 A033181

Keyword

nonn

Author

David W. Wilson

Status

approved