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A347457 Heinz numbers of integer partitions with integer alternating product.

%I A347457 #8 Oct 04 2021 09:26:46
%S A347457 1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,
%T A347457 28,29,31,32,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,52,53,54,
%U A347457 56,57,58,59,61,62,63,64,65,67,68,71,72,73,74,75,76,78
%N A347457 Heinz numbers of integer partitions with integer alternating product.
%C A347457 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 A347457 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347457 Also numbers whose multiset of prime indices has integer reverse-alternating product.
%e A347457 The prime indices of 525 are {2,3,3,4}, with reverse-alternating product 2, so 525 is in the sequence
%e A347457 The prime indices of 135 are {2,2,2,3}, with reverse-alternating product 3/2, so 135 is not in the sequence.
%t A347457 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347457 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347457 Select[Range[100],IntegerQ[altprod[Reverse[primeMS[#]]]]&]
%Y A347457 The reciprocal version is A028982.
%Y A347457 Allowing any alternating product > 1 gives A028983, reverse A347465.
%Y A347457 Factorizations of this type are counted by A347437.
%Y A347457 These partitions are counted by A347446.
%Y A347457 The reverse reciprocal version A347451.
%Y A347457 The odd-length case is A347453.
%Y A347457 The reverse version is A347454.
%Y A347457 The complement is A347455.
%Y A347457 A056239 adds up prime indices, row sums of A112798.
%Y A347457 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347457 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347457 A347461 counts possible alternating products of partitions, reverse A347462.
%Y A347457 Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450.
%K A347457 nonn
%O A347457 1,2
%A A347457 _Gus Wiseman_, Sep 26 2021