A370805 - OEIS

%I #9 May 24 2024 03:51:59

%S 1,0,1,1,2,2,3,4,6,6,9,11,15,18,22,27,34,41,51,62,75,90,109,129,153,

%T 185,217,258,307,359,421

%N Number of condensed integer partitions of n into parts > 1.

%C These are partitions without ones such that it is possible to choose a different divisor of each part.

%e The a(0) = 1 through a(9) = 6 partitions:

%e   ()  .  (2)  (3)  (4)    (5)    (6)    (7)      (8)      (9)

%e                    (2,2)  (3,2)  (3,3)  (4,3)    (4,4)    (5,4)

%e                                  (4,2)  (5,2)    (5,3)    (6,3)

%e                                         (3,2,2)  (6,2)    (7,2)

%e                                                  (3,3,2)  (4,3,2)

%e                                                  (4,2,2)  (5,2,2)

%t Table[Length[Select[IntegerPartitions[n],FreeQ[#,1] && Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]],{n,0,30}]

%Y The version with ones is A239312, complement A370320.

%Y These partitions have as ranks the odd terms of A368110, complement A355740.

%Y The version for prime factors is A370592, complement A370593, post A370807.

%Y The complement without ones is A370804, ranked by the odd terms of A355740.

%Y The version for factorizations is A370814, complement A370813.

%Y A000005 counts divisors.

%Y A000041 counts integer partitions, strict A000009.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Mar 04 2024