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A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

%I A347453 #7 Sep 27 2021 07:56:25
%S A347453 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 A347453 47,48,50,52,53,59,61,63,67,68,71,72,73,75,76,78,79,80,83,89,92,97,98,
%U A347453 99,101,103,107,108,109,112,113,114,116,117,124,125,127,128,130
%N A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.
%C A347453 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 A347453 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347453 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 A347453 The terms and their prime indices begin:
%e A347453       2: {1}         29: {10}            61: {18}
%e A347453       3: {2}         31: {11}            63: {2,2,4}
%e A347453       5: {3}         32: {1,1,1,1,1}     67: {19}
%e A347453       7: {4}         37: {12}            68: {1,1,7}
%e A347453       8: {1,1,1}     41: {13}            71: {20}
%e A347453      11: {5}         42: {1,2,4}         72: {1,1,1,2,2}
%e A347453      12: {1,1,2}     43: {14}            73: {21}
%e A347453      13: {6}         44: {1,1,5}         75: {2,3,3}
%e A347453      17: {7}         45: {2,2,3}         76: {1,1,8}
%e A347453      18: {1,2,2}     47: {15}            78: {1,2,6}
%e A347453      19: {8}         48: {1,1,1,1,2}     79: {22}
%e A347453      20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}
%e A347453      23: {9}         52: {1,1,6}         83: {23}
%e A347453      27: {2,2,2}     53: {16}            89: {24}
%e A347453      28: {1,1,4}     59: {17}            92: {1,1,9}
%t A347453 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347453 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347453 Select[Range[100],OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]
%Y A347453 The reciprocal version is A000290.
%Y A347453 Allowing any alternating product <= 1 gives A001105.
%Y A347453 Allowing any alternating product gives A026424.
%Y A347453 Factorizations of this type are counted by A347441.
%Y A347453 These partitions are counted by A347444.
%Y A347453 Allowing any length gives A347454.
%Y A347453 Allowing any alternating product > 1 gives A347465.
%Y A347453 A027193 counts odd-length partitions.
%Y A347453 A056239 adds up prime indices, row sums of A112798.
%Y A347453 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347453 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347453 A344606 counts alternating permutations of prime indices.
%Y A347453 A347446 counts partitions with integer alternating product.
%Y A347453 A347457 ranks partitions with integer alt product, complement A347455.
%Y A347453 A347461 counts possible alternating products of partitions.
%Y A347453 A347462 counts possible reverse-alternating products of partitions.
%Y A347453 Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.
%K A347453 nonn
%O A347453 1,1
%A A347453 _Gus Wiseman_, Sep 24 2021