login
A338317
Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.
3
1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376
OFFSET
0,5
FORMULA
The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
2 3 4 5 6 7 8 9 A B C
22 32 33 43 44 54 55 65 66
222 52 53 72 73 74 75
322 332 333 433 83 444
2222 522 532 92 543
3222 3322 443 552
22222 533 732
722 3333
3332 5322
5222 33222
32222 222222
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[#, 1]&&(SameQ@@#||CoprimeQ@@Union[#])&]], {n, 0, 15}]
CROSSREFS
A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Sequence in context: A066639 A370808 A111212 * A141286 A165686 A025209
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 24 2020
STATUS
approved