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A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

%I A347451 #5 Sep 27 2021 07:56:16
%S A347451 1,2,4,6,8,9,10,14,16,18,21,22,24,25,26,32,34,36,38,39,40,46,49,50,54,
%T A347451 56,57,58,62,64,65,72,74,81,82,84,86,87,88,90,94,96,98,100,104,106,
%U A347451 111,115,118,121,122,126,128,129,133,134,136,142,144,146,150,152
%N A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.
%C A347451 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 A347451 We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
%C A347451 Also Heinz numbers integer partitions with integer reverse-reciprocal alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A347451 The terms and their prime indices begin:
%e A347451       1: {}            32: {1,1,1,1,1}       65: {3,6}
%e A347451       2: {1}           34: {1,7}             72: {1,1,1,2,2}
%e A347451       4: {1,1}         36: {1,1,2,2}         74: {1,12}
%e A347451       6: {1,2}         38: {1,8}             81: {2,2,2,2}
%e A347451       8: {1,1,1}       39: {2,6}             82: {1,13}
%e A347451       9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
%e A347451      10: {1,3}         46: {1,9}             86: {1,14}
%e A347451      14: {1,4}         49: {4,4}             87: {2,10}
%e A347451      16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
%e A347451      18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
%e A347451      21: {2,4}         56: {1,1,1,4}         94: {1,15}
%e A347451      22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
%e A347451      24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
%e A347451      25: {3,3}         62: {1,11}           100: {1,1,3,3}
%e A347451      26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}
%t A347451 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347451 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347451 Select[Range[100],IntegerQ[1/altprod[primeMS[#]]]&]
%Y A347451 The version for reversed prime indices is A028982, counted by A119620.
%Y A347451 The additive version is A119899, strict A028260.
%Y A347451 Allowing any alternating product >= 1 gives A344609.
%Y A347451 Factorizations of this type are counted by A347439.
%Y A347451 Allowing any alternating product <= 1 gives A347450.
%Y A347451 The non-reciprocal version is A347454.
%Y A347451 Allowing any alternating product > 1 gives A347465, reverse A028983.
%Y A347451 A056239 adds up prime indices, row sums of A112798.
%Y A347451 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347451 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347451 A344606 counts alternating permutations of prime indices.
%Y A347451 A347457 ranks partitions with integer alternating product.
%Y A347451 Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.
%K A347451 nonn
%O A347451 1,2
%A A347451 _Gus Wiseman_, Sep 24 2021