A284613 -id:A284613 - cp's OEIS frontend

A284612 Number of partitions of n such that the (sum of distinct odd parts) < n/2.

0, 1, 2, 4, 4, 7, 9, 13, 22, 31, 40, 53, 75, 98, 126, 156, 215, 272, 375, 461, 602, 743, 971, 1175, 1491, 1817, 2359, 2831, 3499, 4210, 5326, 6329, 7956, 9501, 11833, 13994, 17155, 20324, 24845, 29189, 35688, 42016, 50803, 59379, 71893, 84106, 100369, 116691

Offset: 1

Clark Kimberling, Mar 30 2017

			a(6) counts these 7 partitions: 6, 41, 411, 222, 2211, 21111, 111111.

Cf. A284613, A284614, A284615.

Table[p = IntegerPartitions[n];
Length[Select[Table[Total[Select[DeleteDuplicates[p[[k]]], OddQ]], {k, Length[p]}], # < n/2 &]], {n, 55}]

A284614 Number of partitions of n such that the (sum of distinct odd parts) > n/2.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 6, 4, 8, 10, 16, 16, 26, 36, 50, 55, 82, 85, 115, 136, 190, 216, 284, 340, 467, 500, 651, 801, 1066, 1181, 1516, 1665, 2187, 2393, 3050, 3466, 4482, 5028, 6340, 6951, 8895, 9953, 12458, 13640, 17241, 19649, 24385, 26386, 33078, 36138, 44569

Offset: 1

Clark Kimberling, Apr 02 2017

			a(5) counts these 3 partitions: 5, 32, 311.

Cf. A284612, A284613, A284615.

Table[p = IntegerPartitions[n];
 Length[Select[Table[Total[Select[DeleteDuplicates[p[[k]]], OddQ]], {k, Length[p]}], # > n/2 &]], {n, 55}]

A284615 Number of partitions of n such that the (sum of distinct odd parts) >= n/2.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 6, 9, 8, 11, 16, 24, 26, 37, 50, 75, 82, 113, 115, 166, 190, 259, 284, 400, 467, 619, 651, 887, 1066, 1394, 1516, 2020, 2187, 2809, 3050, 3983, 4482, 5691, 6340, 8149, 8895, 11158, 12458, 15796, 17241, 21452, 24385, 30582, 33078, 40775

Offset: 1

Clark Kimberling, Apr 02 2017

Cf. A284612, A284613, A284614.

Table[p = IntegerPartitions[n];
 Length[Select[Table[Total[Select[DeleteDuplicates[p[[k]]], OddQ]], {k, Length[p]}], # >= n/2 &]], {n, 55}]

a(5) counts these 3 partitions: 5, 32, 311.

A284608 Number of partitions of n such that the (sum of distinct odd parts) = floor(n/2).

Original entry on oeis.org

0, 1, 2, 0, 0, 1, 2, 5, 6, 1, 3, 8, 9, 1, 5, 20, 20, 28, 45, 30, 29, 43, 69, 60, 51, 119, 174, 86, 75, 213, 307, 355, 375, 416, 583, 517, 541, 663, 923, 1198, 1291, 1205, 1650, 2156, 2365, 1803, 2469, 4196, 4596, 4637, 6073, 6684, 7374, 6740, 8829, 12345

Offset: 1

Clark Kimberling, Mar 30 2017

			a(8) counts these 5 partitions: 431, 3311, 3221, 32111, 311111.

Cf. A284609, A284610, A284612, A284613.

Table[p = IntegerPartitions[n]; Length[Select[Table[Total[Select[DeleteDuplicates[p[[k]]], OddQ]], {k, Length[p]}], # == Floor[n/2] &]], {n, 60}] (* Peter J. C. Moses, Mar 29 2017 *)

cp's OEIS Frontend

A284612 Number of partitions of n such that the (sum of distinct odd parts) < n/2.

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A284614 Number of partitions of n such that the (sum of distinct odd parts) > n/2.

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A284615 Number of partitions of n such that the (sum of distinct odd parts) >= n/2.

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A284608 Number of partitions of n such that the (sum of distinct odd parts) = floor(n/2).

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