A370808 - OEIS

%I #12 Sep 17 2024 08:44:38

%S 1,1,2,2,3,4,5,6,7,10,11,14,17,19,23,29,30,39,41,51,58,66,78,82,102,

%T 110,132,144,162,186,210,228,260,296,328,366,412,462,512,560,638,692,

%U 764,860,924,1028,1122,1276,1406,1528,1721,1898,2056,2318,2506,2812,3020,3442

%N Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

%e For the partitions of 5 we have the following choices:

%e       (5): {{1},{5}}

%e      (41): {{1,1},{1,2},{1,4}}

%e      (32): {{1,1},{1,2},{1,3},{2,3}}

%e     (311): {{1,1,1},{1,1,3}}

%e     (221): {{1,1,1},{1,1,2},{1,2,2}}

%e    (2111): {{1,1,1,1},{1,1,1,2}}

%e   (11111): {{1,1,1,1,1}}

%e So a(5) = 4.

%t Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@IntegerPartitions[n]],{n,0,30}]

%Y For just prime factors we have A370809.

%Y The version for factorizations is A370816, for just prime factors A370817.

%Y A000005 counts divisors.

%Y A000041 counts integer partitions, strict A000009.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A239312 counts condensed partitions, ranks A368110.

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

%Y A355733 counts choices of divisors of prime indicec.

%Y A370320 counts non-condensed partitions, ranks A355740.

%Y A370592 counts factor-choosable partitions, complement A370593.

%Y Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 05 2024

%E Terms a(31) onward from _Max Alekseyev_, Sep 17 2024