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A350486
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Numbers that have an equal number of even- and odd-length unordered factorizations and also an equal number of even- and odd-length unordered factorizations into distinct factors.
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0
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6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 180, 183, 185, 187, 192, 194
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OFFSET
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1,1
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COMMENTS
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First differs from A006881 at a(53) = 180.
By length, we mean the number of factors in a particular factorization.
Intersection of A319240 (factors are not necessarily distinct) and A319238 (factors are distinct).
There are infinitely many terms in this sequence since all squarefree semiprimes (listed in A006881) are always such numbers.
There are no terms of the form p^k with p prime (listed in A000961).
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures.
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LINKS
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EXAMPLE
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6=2*3 (unrestricted) has an equal number (1) of even-length factorizations and odd-length factorizations, and 6=2*3 (distinct) has an equal number (1) of even-length factorizations and odd-length factorizations.
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MATHEMATICA
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facs[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[facs[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Intersection @@ First@Flatten[Position[#, 0] & /@ Transpose@Table[Sum[(-1)^Length[f], {f, #}] & /@ {facs[n], Select[facs[n], UnsameQ @@ # &]}, {n, #1, #2}], {3}]&[1, 194] (* Robert P. P. McKone, Jan 05 2022, from Gus Wiseman in A319238 and A319240 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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