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A347450
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Numbers whose multiset of prime indices has alternating product <= 1.
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17
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1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 32, 33, 34, 35, 36, 38, 39, 40, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 72, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 100, 104, 106, 111, 115, 118, 119, 121, 122
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OFFSET
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1,2
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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)).
Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers whose multiset of prime indices has alternating sum <= 1.
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LINKS
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FORMULA
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EXAMPLE
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The initial terms and their prime indices:
1: {} 26: {1,6} 56: {1,1,1,4}
2: {1} 32: {1,1,1,1,1} 57: {2,8}
4: {1,1} 33: {2,5} 58: {1,10}
6: {1,2} 34: {1,7} 60: {1,1,2,3}
8: {1,1,1} 35: {3,4} 62: {1,11}
9: {2,2} 36: {1,1,2,2} 64: {1,1,1,1,1,1}
10: {1,3} 38: {1,8} 65: {3,6}
14: {1,4} 39: {2,6} 69: {2,9}
15: {2,3} 40: {1,1,1,3} 72: {1,1,1,2,2}
16: {1,1,1,1} 46: {1,9} 74: {1,12}
18: {1,2,2} 49: {4,4} 77: {4,5}
21: {2,4} 50: {1,3,3} 81: {2,2,2,2}
22: {1,5} 51: {2,7} 82: {1,13}
24: {1,1,1,2} 54: {1,2,2,2} 84: {1,1,2,4}
25: {3,3} 55: {3,5} 85: {3,7}
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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 additive version (alternating sum <= 0) is A028260.
Allowing any alternating product < 1 gives A119899.
Factorizations of this type are counted by A339846, complement A339890.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Partitions of this type are counted by A347443.
Allowing any integer alternating product gives A347454, reciprocal A347451.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347457 lists Heinz numbers of partitions with integer alternating product.
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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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