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A347465
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Numbers whose multiset of prime indices has alternating product > 1.
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10
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3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127
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OFFSET
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1,1
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COMMENTS
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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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.
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LINKS
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EXAMPLE
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The terms and their prime indices begin:
3: {2} 37: {12} 68: {1,1,7}
5: {3} 41: {13} 70: {1,3,4}
7: {4} 42: {1,2,4} 71: {20}
11: {5} 43: {14} 73: {21}
12: {1,1,2} 44: {1,1,5} 75: {2,3,3}
13: {6} 45: {2,2,3} 76: {1,1,8}
17: {7} 47: {15} 78: {1,2,6}
19: {8} 48: {1,1,1,1,2} 79: {22}
20: {1,1,3} 52: {1,1,6} 80: {1,1,1,1,3}
23: {9} 53: {16} 83: {23}
27: {2,2,2} 59: {17} 89: {24}
28: {1,1,4} 61: {18} 92: {1,1,9}
29: {10} 63: {2,2,4} 97: {25}
30: {1,2,3} 66: {1,2,5} 99: {2,2,5}
31: {11} 67: {19} 101: {26}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Select[Range[100], altprod[primeMS[#]]>1&]
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CROSSREFS
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The squarefree case is A030059 without 2.
The opposite version (< 1 instead of > 1) is A119899.
The weak version (>= 1 instead of > 1) is A344609.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A347446 counts partitions with integer alternating product, reverse A347445.
Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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