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A347454 Numbers whose multiset of prime indices has integer alternating product.

%I A347454 #8 Sep 27 2021 07:56:32
%S A347454 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,25,27,28,29,31,32,36,37,
%T A347454 41,42,43,44,45,47,48,49,50,52,53,59,61,63,64,67,68,71,72,73,75,76,78,
%U A347454 79,80,81,83,89,92,97,98,99,100,101,103,107,108,109,112,113
%N A347454 Numbers whose multiset of prime indices has integer alternating product.
%C A347454 First differs from A265640 in having 42.
%C A347454 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A347454 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347454 Also Heinz numbers of partitions with integer reverse-alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A347454 The terms and their prime indices begin:
%e A347454       1: {}            20: {1,1,3}         47: {15}
%e A347454       2: {1}           23: {9}             48: {1,1,1,1,2}
%e A347454       3: {2}           25: {3,3}           49: {4,4}
%e A347454       4: {1,1}         27: {2,2,2}         50: {1,3,3}
%e A347454       5: {3}           28: {1,1,4}         52: {1,1,6}
%e A347454       7: {4}           29: {10}            53: {16}
%e A347454       8: {1,1,1}       31: {11}            59: {17}
%e A347454       9: {2,2}         32: {1,1,1,1,1}     61: {18}
%e A347454      11: {5}           36: {1,1,2,2}       63: {2,2,4}
%e A347454      12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
%e A347454      13: {6}           41: {13}            67: {19}
%e A347454      16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
%e A347454      17: {7}           43: {14}            71: {20}
%e A347454      18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
%e A347454      19: {8}           45: {2,2,3}         73: {21}
%t A347454 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347454 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347454 Select[Range[100],IntegerQ[altprod[primeMS[#]]]&]
%Y A347454 The even-length case is A000290.
%Y A347454 The additive version is A026424.
%Y A347454 Allowing any alternating product < 1 gives A119899, strict A028260.
%Y A347454 Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
%Y A347454 Factorizations of this type are counted by A347437.
%Y A347454 These partitions are counted by A347445, reverse A347446.
%Y A347454 Allowing any alternating product <= 1 gives A347450.
%Y A347454 The reciprocal version is A347451.
%Y A347454 The odd-length case is A347453.
%Y A347454 The version for reversed prime indices is A347457, complement A347455.
%Y A347454 Allowing any alternating product > 1 gives A347465, reverse A028983.
%Y A347454 A056239 adds up prime indices, row sums of A112798.
%Y A347454 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347454 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347454 A344606 counts alternating permutations of prime indices.
%Y A347454 A347461 counts possible alternating products of partitions.
%Y A347454 A347462 counts possible reverse-alternating products of partitions.
%Y A347454 Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.
%K A347454 nonn
%O A347454 1,2
%A A347454 _Gus Wiseman_, Sep 26 2021