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%I #6 Dec 10 2025 16:31:58
%S 1,0,1,1,1,2,1,3,2,4,3,7,5,8,8,12,11,16,17,23,24,30,33,42,46,54,62,75,
%T 83,98,111,130,146,168,192,222,249,284,322,368,413,470,529,599,673,
%U 758,851,961,1074,1204,1348,1511,1685,1884,2100,2344,2606,2902,3225
%N Number of strict integer partitions of n > 0 such that the least part and the greatest part are both odd (equivalently, their product is odd).
%e The a(8) = 3 through a(14) = 8 partitions:
%e (5,3) (9) (7,3) (11) (7,5) (13) (9,5)
%e (7,1) (5,3,1) (9,1) (7,3,1) (9,3) (7,5,1) (11,3)
%e (5,2,1) (5,4,1) (5,3,2,1) (11,1) (9,3,1) (13,1)
%e (7,2,1) (5,4,3) (5,4,3,1) (7,4,3)
%e (7,4,1) (7,3,2,1) (7,6,1)
%e (9,2,1) (9,4,1)
%e (5,4,2,1) (11,2,1)
%e (7,4,2,1)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[First[#]*Last[#]]&]],{n,30}]
%Y For odd least part we have A026832, non-strict A026804, ranks A340932.
%Y For even least part we have A026833, non-strict A026805, ranks A340933.
%Y For odd greatest part we have A026837, non-strict A027193, ranks A244991.
%Y For even greatest part we have A067661, non-strict A027187, ranks A244990.
%Y The non-strict version is A325338, ranks A390093.
%Y For sum instead of product we have A390746, non-strict A390092, ranks A390988.
%Y The complement is counted by A391231, non-strict A391230, ranks A391229.
%Y For both parts even we have A391233, non-strict A325346, ranks A391232.
%Y A000041 counts integer partitions, strict A000009.
%Y A006141 counts partitions with least part = length, ranks A324522.
%Y A257991 counts odd prime indices, even A257992.
%Y A333352 gives least prime index times greatest prime index.
%Y Cf. A005408, A035457, A058695, A066208, A067659, A101707, A160786, A300063, A300272, A340832, A391227.
%K nonn
%O 1,6
%A _Gus Wiseman_, Dec 10 2025