

A337976


Number of partitions of n into two distinct parts (s,t), such that s  t, (ts)  n, and where n/(ts) <= s < t.


1



0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1
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OFFSET

1,12


LINKS

Table of n, a(n) for n=1..93.


FORMULA

a(n) = Sum_{i=1..floor((n1)/2)} Sum_{k=1..i} [n = k*(n2*i)] * (1  ceiling((ni)/i) + floor((ni)/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 (62)  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 (63)  9 where 9/3 = 3 <= 3, so a(9) = 1.
a(10) = 0; The partition (6,4) has difference of (64) = 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 93 = 6, 84 = 4 and 75 = 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

Cf. A337509 (same without s  t).
Sequence in context: A101606 A257469 A275947 * A125005 A309367 A122179
Adjacent sequences: A337973 A337974 A337975 * A337977 A337978 A337979


KEYWORD

nonn


AUTHOR

Wesley Ivan Hurt, Oct 05 2020


STATUS

approved



