A349150 - OEIS

%I #7 Nov 13 2021 10:22:44

%S 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 41,42,43,45,47,49,51,53,54,57,58,59,61,63,65,67,69,71,73,74,77,78,79,

%U 81,83,86,87,89,91,93,95,97,98,99,101,103,105,106,107,109

%N Heinz numbers of integer partitions with at most 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 at most one odd prime index.

%C Also Heinz numbers of partitions with conjugate alternating sum <= 1.

%F Union of A066207 (no odd parts) and A349158 (one odd part).

%e The terms and their prime indices begin:

%e       1: {}          23: {9}         49: {4,4}

%e       2: {1}         26: {1,6}       51: {2,7}

%e       3: {2}         27: {2,2,2}     53: {16}

%e       5: {3}         29: {10}        54: {1,2,2,2}

%e       6: {1,2}       31: {11}        57: {2,8}

%e       7: {4}         33: {2,5}       58: {1,10}

%e       9: {2,2}       35: {3,4}       59: {17}

%e      11: {5}         37: {12}        61: {18}

%e      13: {6}         38: {1,8}       63: {2,2,4}

%e      14: {1,4}       39: {2,6}       65: {3,6}

%e      15: {2,3}       41: {13}        67: {19}

%e      17: {7}         42: {1,2,4}     69: {2,9}

%e      18: {1,2,2}     43: {14}        71: {20}

%e      19: {8}         45: {2,2,3}     73: {21}

%e      21: {2,4}       47: {15}        74: {1,12}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]

%Y The case of no odd parts is A066207, counted by A000041 up to 0's.

%Y Requiring all odd parts gives A066208, counted by A000009.

%Y These partitions are counted by A100824, even-length case A349149.

%Y These are the positions of 0's and 1's in A257991.

%Y The conjugate partitions are ranked by A349151.

%Y The case of one odd part is A349158, counted by A000070 up to 0's.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A122111 is a representation of partition conjugation.

%Y A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.

%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 A340932 ranks partitions whose least part is odd, counted by A026804.

%Y A345958 ranks partitions with alternating sum 1.

%Y A349157 ranks partitions with as many even parts as odd conjugate parts.

%Y Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 10 2021