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).

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

Links

Table of n, a(n) for n=2..80.

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

Cf. A039653, A083207.

Sequence in context: A358940 A136315 A011159 * A091556 A299207 A277330

Adjacent sequences: A359620 A359621 A359622 * A359624 A359625 A359626

Keyword

nonn

Author

Rémy Sigrist, Feb 17 2023

Status

approved