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A349150 Heinz numbers of integer partitions with at most one odd part.

%I A349150 #7 Nov 13 2021 10:22:44
%S A349150 1,2,3,5,6,7,9,11,13,14,15,17,18,19,21,23,26,27,29,31,33,35,37,38,39,
%T A349150 41,42,43,45,47,49,51,53,54,57,58,59,61,63,65,67,69,71,73,74,77,78,79,
%U A349150 81,83,86,87,89,91,93,95,97,98,99,101,103,105,106,107,109
%N A349150 Heinz numbers of integer partitions with at most one odd part.
%C A349150 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.
%C A349150 Also Heinz numbers of partitions with conjugate alternating sum <= 1.
%F A349150 Union of A066207 (no odd parts) and A349158 (one odd part).
%e A349150 The terms and their prime indices begin:
%e A349150       1: {}          23: {9}         49: {4,4}
%e A349150       2: {1}         26: {1,6}       51: {2,7}
%e A349150       3: {2}         27: {2,2,2}     53: {16}
%e A349150       5: {3}         29: {10}        54: {1,2,2,2}
%e A349150       6: {1,2}       31: {11}        57: {2,8}
%e A349150       7: {4}         33: {2,5}       58: {1,10}
%e A349150       9: {2,2}       35: {3,4}       59: {17}
%e A349150      11: {5}         37: {12}        61: {18}
%e A349150      13: {6}         38: {1,8}       63: {2,2,4}
%e A349150      14: {1,4}       39: {2,6}       65: {3,6}
%e A349150      15: {2,3}       41: {13}        67: {19}
%e A349150      17: {7}         42: {1,2,4}     69: {2,9}
%e A349150      18: {1,2,2}     43: {14}        71: {20}
%e A349150      19: {8}         45: {2,2,3}     73: {21}
%e A349150      21: {2,4}       47: {15}        74: {1,12}
%t A349150 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349150 Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]
%Y A349150 The case of no odd parts is A066207, counted by A000041 up to 0's.
%Y A349150 Requiring all odd parts gives A066208, counted by A000009.
%Y A349150 These partitions are counted by A100824, even-length case A349149.
%Y A349150 These are the positions of 0's and 1's in A257991.
%Y A349150 The conjugate partitions are ranked by A349151.
%Y A349150 The case of one odd part is A349158, counted by A000070 up to 0's.
%Y A349150 A056239 adds up prime indices, row sums of A112798.
%Y A349150 A122111 is a representation of partition conjugation.
%Y A349150 A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.
%Y A349150 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A349150 A325698 ranks partitions with as many even as odd parts, counted by A045931.
%Y A349150 A340932 ranks partitions whose least part is odd, counted by A026804.
%Y A349150 A345958 ranks partitions with alternating sum 1.
%Y A349150 A349157 ranks partitions with as many even parts as odd conjugate parts.
%Y A349150 Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.
%K A349150 nonn
%O A349150 1,2
%A A349150 _Gus Wiseman_, Nov 10 2021