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%I #8 Dec 04 2021 12:38:02
%S 2,5,6,11,14,15,17,18,23,26,31,33,35,38,41,42,45,47,51,54,58,59,65,67,
%T 69,73,74,77,78,83,86,93,95,97,98,99,103,105,106,109,114,119,122,123,
%U 126,127,135,137,141,142,143,145,149,153,157,158,161,162,167,174
%N Heinz numbers of integer partitions with exactly one odd part.
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with exactly one odd prime index. These are also partitions whose conjugate partition has alternating sum equal to 1.
%C Numbers that are product of a term of A031368 and a term of A066207. - _Antti Karttunen_, Nov 13 2021
%e The terms and corresponding partitions begin:
%e 2: (1) 42: (4,2,1) 86: (14,1)
%e 5: (3) 45: (3,2,2) 93: (11,2)
%e 6: (2,1) 47: (15) 95: (8,3)
%e 11: (5) 51: (7,2) 97: (25)
%e 14: (4,1) 54: (2,2,2,1) 98: (4,4,1)
%e 15: (3,2) 58: (10,1) 99: (5,2,2)
%e 17: (7) 59: (17) 103: (27)
%e 18: (2,2,1) 65: (6,3) 105: (4,3,2)
%e 23: (9) 67: (19) 106: (16,1)
%e 26: (6,1) 69: (9,2) 109: (29)
%e 31: (11) 73: (21) 114: (8,2,1)
%e 33: (5,2) 74: (12,1) 119: (7,4)
%e 35: (4,3) 77: (5,4) 122: (18,1)
%e 38: (8,1) 78: (6,2,1) 123: (13,2)
%e 41: (13) 83: (23) 126: (4,2,2,1)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Count[primeMS[#],_?OddQ]==1&]
%Y These partitions are counted by A000070 up to 0's.
%Y Allowing no odd parts gives A066207, counted by A000041 up to 0's.
%Y Requiring all odd parts gives A066208, counted by A000009.
%Y These are the positions of 1's in A257991.
%Y The even prime indices are counted by A257992.
%Y The conjugate partitions are ranked by A345958.
%Y Allowing at most one odd part gives A349150, counted by A100824.
%Y A047993 ranks balanced partitions, counted by A106529.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A122111 is a representation of partition conjugation.
%Y A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A325698 ranks partitions with as many even as odd parts, counted by A045931.
%Y A340604 ranks partitions of odd positive rank, counted by A101707.
%Y A340932 ranks partitions whose least part is odd, counted by A026804.
%Y A349157 ranks partitions with as many even parts as odd conjugate parts.
%Y Cf. A000700, A001222, A027187, A027193, A028260, A031368 (primes with odd index), A035363, A215366, A277579, A300063, A349151.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 12 2021