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A349151 Heinz numbers of integer partitions with alternating sum <= 1.

%I A349151 #6 Nov 13 2021 10:22:50
%S A349151 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 A349151 100,121,128,135,140,143,144,150,162,169,196,200,216,221,225,240,242,
%U A349151 256,288,289,294,308,315,323,324,338,361,375,384,392,400,437,441,450
%N A349151 Heinz numbers of integer partitions with alternating sum <= 1.
%C A349151 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 A349151 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 A349151 Equals A000290 \/ A345958 decapitated.
%e A349151 The terms and their prime indices begin:
%e A349151     1: {}
%e A349151     2: {1}
%e A349151     4: {1,1}
%e A349151     6: {1,2}
%e A349151     8: {1,1,1}
%e A349151     9: {2,2}
%e A349151    15: {2,3}
%e A349151    16: {1,1,1,1}
%e A349151    18: {1,2,2}
%e A349151    24: {1,1,1,2}
%e A349151    25: {3,3}
%e A349151    32: {1,1,1,1,1}
%e A349151    35: {3,4}
%e A349151    36: {1,1,2,2}
%e A349151    49: {4,4}
%t A349151 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349151 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A349151 Select[Range[100],ats[Reverse[primeMS[#]]]<=1&]
%Y A349151 The case of alternating sum 0 is A000290.
%Y A349151 These partitions are counted by A100824.
%Y A349151 These are the positions of 0's and 1's in A344616.
%Y A349151 The case of alternating sum 1 is A345958.
%Y A349151 The conjugate partitions are ranked by A349150.
%Y A349151 A000041 counts integer partitions.
%Y A349151 A056239 adds up prime indices, row sums of A112798.
%Y A349151 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A349151 A106529 ranks balanced partitions, counted by A047993.
%Y A349151 A122111 is a representation of partition conjugation.
%Y A349151 A257991 counts odd prime indices.
%Y A349151 A316524 gives the alternating sum of prime indices.
%Y A349151 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A349151 A349157 ranks partitions with as many even parts as odd conjugate parts.
%Y A349151 Cf. A000070, A000700, A001222, A027187, A027193, A215366, A277103, A277579, A326841, A349149, A349158.
%K A349151 nonn
%O A349151 1,2
%A A349151 _Gus Wiseman_, Nov 10 2021