A349151 - OEIS

%I #6 Nov 13 2021 10:22:50

%S 1,2,4,6,8,9,15,16,18,24,25,32,35,36,49,50,54,60,64,72,77,81,96,98,

%T 100,121,128,135,140,143,144,150,162,169,196,200,216,221,225,240,242,

%U 256,288,289,294,308,315,323,324,338,361,375,384,392,400,437,441,450

%N Heinz numbers of integer partitions with alternating sum <= 1.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so these are also Heinz numbers of partitions with at most one odd conjugate part.

%F Equals A000290 \/ A345958 decapitated.

%e The terms and their prime indices begin:

%e     1: {}

%e     2: {1}

%e     4: {1,1}

%e     6: {1,2}

%e     8: {1,1,1}

%e     9: {2,2}

%e    15: {2,3}

%e    16: {1,1,1,1}

%e    18: {1,2,2}

%e    24: {1,1,1,2}

%e    25: {3,3}

%e    32: {1,1,1,1,1}

%e    35: {3,4}

%e    36: {1,1,2,2}

%e    49: {4,4}

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

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Select[Range[100],ats[Reverse[primeMS[#]]]<=1&]

%Y The case of alternating sum 0 is A000290.

%Y These partitions are counted by A100824.

%Y These are the positions of 0's and 1's in A344616.

%Y The case of alternating sum 1 is A345958.

%Y The conjugate partitions are ranked by A349150.

%Y A000041 counts integer partitions.

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

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A106529 ranks balanced partitions, counted by A047993.

%Y A122111 is a representation of partition conjugation.

%Y A257991 counts odd prime indices.

%Y A316524 gives the alternating sum of prime indices.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

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

%Y Cf. A000070, A000700, A001222, A027187, A027193, A215366, A277103, A277579, A326841, A349149, A349158.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 10 2021