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A359623
a(n) is the least integer of the form sum(X)/sum(Y) where {X, Y} runs through the partitions of the divisors of n into two nonempty sets (and sum(Z) is the sum of the elements of Z).
0
2, 3, 6, 5, 1, 7, 2, 12, 2, 11, 1, 13, 2, 2, 30, 17, 2, 19, 1, 3, 2, 23, 1, 30, 2, 3, 1, 29, 1, 31, 2, 3, 2, 3, 6, 37, 2, 3, 1, 41, 1, 43, 2, 5, 2, 47, 1, 56, 2, 3, 6, 53, 1, 5, 1, 3, 2, 59, 1, 61, 2, 7, 126, 5, 1, 67, 5, 3, 1, 71, 2, 73, 2, 3, 6, 7, 1, 79, 1
OFFSET
2,1
FORMULA
a(n) <= A039653(n).
a(n) = 1 iff n is a Zumkeller number (A083207).
a(n) = n iff n is a prime number.
EXAMPLE
The first terms, alongside a suitable partition, are:
n a(n) Partition
-- ---- ----------------------
2 2 {2}, {1}
3 3 {3}, {1}
4 6 {2, 4}, {1}
5 5 {5}, {1}
6 1 {1, 2, 3}, {6}
7 7 {7}, {1}
8 2 {2, 8}, {1, 4}
9 12 {3, 9}, {1}
10 2 {2, 10}, {1, 5}
11 11 {11}, {1}
12 1 {1, 3, 4, 6}, {2, 12}
13 13 {13}, {1}
14 2 {2, 14}, {1, 7}
15 2 {1, 15}, {3, 5}
PROG
(PARI) a(n) = { my (d=divisors(n), t=vecsum(d), s=[0]); for (i=1, #d, s = setunion(s, [v+d[i] | v <- s]); ); for (i=2, oo, my (r=s[i]/(t-s[i])); if (denominator(r)==1, return (r))) }
CROSSREFS
Sequence in context: A358940 A136315 A011159 * A091556 A299207 A277330
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Feb 17 2023
STATUS
approved