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%I #7 Sep 27 2021 07:56:25
%S 2,3,5,7,8,11,12,13,17,18,19,20,23,27,28,29,31,32,37,41,42,43,44,45,
%T 47,48,50,52,53,59,61,63,67,68,71,72,73,75,76,78,79,80,83,89,92,97,98,
%U 99,101,103,107,108,109,112,113,114,116,117,124,125,127,128,130
%N Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.
%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 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C Also numbers whose multiset of prime indices has odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.
%e The terms and their prime indices begin:
%e 2: {1} 29: {10} 61: {18}
%e 3: {2} 31: {11} 63: {2,2,4}
%e 5: {3} 32: {1,1,1,1,1} 67: {19}
%e 7: {4} 37: {12} 68: {1,1,7}
%e 8: {1,1,1} 41: {13} 71: {20}
%e 11: {5} 42: {1,2,4} 72: {1,1,1,2,2}
%e 12: {1,1,2} 43: {14} 73: {21}
%e 13: {6} 44: {1,1,5} 75: {2,3,3}
%e 17: {7} 45: {2,2,3} 76: {1,1,8}
%e 18: {1,2,2} 47: {15} 78: {1,2,6}
%e 19: {8} 48: {1,1,1,1,2} 79: {22}
%e 20: {1,1,3} 50: {1,3,3} 80: {1,1,1,1,3}
%e 23: {9} 52: {1,1,6} 83: {23}
%e 27: {2,2,2} 53: {16} 89: {24}
%e 28: {1,1,4} 59: {17} 92: {1,1,9}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t Select[Range[100],OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]
%Y The reciprocal version is A000290.
%Y Allowing any alternating product <= 1 gives A001105.
%Y Allowing any alternating product gives A026424.
%Y Factorizations of this type are counted by A347441.
%Y These partitions are counted by A347444.
%Y Allowing any length gives A347454.
%Y Allowing any alternating product > 1 gives A347465.
%Y A027193 counts odd-length partitions.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A344606 counts alternating permutations of prime indices.
%Y A347446 counts partitions with integer alternating product.
%Y A347457 ranks partitions with integer alt product, complement A347455.
%Y A347461 counts possible alternating products of partitions.
%Y A347462 counts possible reverse-alternating products of partitions.
%Y Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.
%K nonn
%O 1,1
%A _Gus Wiseman_, Sep 24 2021
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