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A341866
The cardinality of the smallest (nontrivial, except for prime n) multiset of positive integers whose product and sum equal n.
1
1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 1, 6, 1, 7, 9, 8, 1, 9, 1, 10, 13, 11, 1, 12, 17, 13, 17, 14, 1, 15, 1, 16, 21, 17, 25, 18, 1, 19, 25, 20, 1, 21, 1, 22, 29, 23, 1, 24, 37, 25, 33, 26, 1, 27, 41, 28, 37, 29, 1, 30, 1, 31, 41, 32
OFFSET
1,4
COMMENTS
The smallest set is obtained by taking the largest such multiset (A341865(n)) and replacing the largest proper subset that is also a product-sum multiset with its product. A singleton would always be the smallest product-sum multiset, so those are excluded except for prime n where no nontrivial multisets exist.
FORMULA
a(n) = (n/p - 1)*(p-1) + 1, where p is the smallest factor of n.
a(n) = A341865(n) - A341865(n/p) + 1, where p is the smallest prime factor of n.
EXAMPLE
For n = 12, the set of size a(n) = 6 is {1,1,1,1,2,6}.
PROG
(PARI) a(n) = if (n==1, 1, my(p=vecmin(factor(n)[, 1])); (n/p-1)*(p-1) + 1); \\ Michel Marcus, Feb 26 2021
CROSSREFS
Equals A330492 + 1. - Hugo Pfoertner, Feb 23 2021
Sequence in context: A142878 A299774 A362254 * A300242 A293892 A295885
KEYWORD
nonn
AUTHOR
Nathaniel Gregg, Feb 22 2021
STATUS
approved