OFFSET
1,12
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} Sum_{k=1..i} [n = k*(n-2*i)] * (1 - ceiling((n-i)/i) + floor((n-i)/i)), where [ ] is the Iverson bracket.
EXAMPLE
a(8) = 1; There are 3 partitions of 8 into two distinct parts: (7,1), (6,2), (5,3), with differences 6, 4 and 2. Only the partition (6,2) satisfies 2 | 6 and (6-2) | 8 where 8/4 = 2 <= 2, so a(8) = 1.
a(9) = 1; There are 4 partitions of 9 into two distinct parts: (8,1), (7,2), (6,3), (5,4) with differences 7, 5, 3 and 1. Only the partition (6,3) satisfies 3 | 6 and (6-3) | 9 where 9/3 = 3 <= 3, so a(9) = 1.
a(10) = 0; The partition (6,4) has difference of (6-4) = 2 | 10, but neither 4 | 6 and 10/2 = 5 > 4. So a(10) = 0.
a(11) = 0; No difference divides 11 (prime), so a(11) = 0.
a(12) = 2; Check (9,3), (8,4) and (7,5) since 9-3 = 6, 8-4 = 4 and 7-5 = 2 all divide 12. Then we have 3 | 9 with 12/6 = 2 < 3 and 4 | 8 with 12/4 = 3 < 4, but for (7,5), 5 does not divide 7 and moreover 12/2 = 6 > 5.
MATHEMATICA
Table[Sum[Sum[KroneckerDelta[k (n - 2 i), n] (1 - Ceiling[(n - i)/i] + Floor[(n - i)/i]), {k, i}], {i, Floor[(n - 1)/2]}], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Oct 05 2020
STATUS
approved