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A279767
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Numbers m such that m and m+2 have the same prime signature.
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2
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3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413
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OFFSET
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1,1
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COMMENTS
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The sequence contains some terms such that m and m + 2k (k > 1) have the same prime signature. For some terms where m and m + 2k share the same prime signature this means that every alternate element between, and including m and m + 2k have the same prime signature. The first such example is where a(41951)=402677, a(41953)=402679, and a(41955)=402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example, a(41952)=402678 and a(41954)=402680 share the prime signature {1,1,3}. - Torlach Rush, Feb 25 2018
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LINKS
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EXAMPLE
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18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.
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MATHEMATICA
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primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)
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PROG
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(PARI) isok(n) = vecsort(factor(n)[, 2]) == vecsort(factor(n+2)[, 2]); \\ Michel Marcus, Feb 25 2018
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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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