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A279767 Numbers m such that m and m+2 have the same prime signature.

%I A279767 #47 May 19 2024 02:13:23
%S A279767 3,5,11,17,18,29,33,41,50,54,55,59,71,85,91,93,101,107,137,141,143,
%T A279767 149,159,179,183,185,191,197,201,203,213,215,217,219,227,235,239,242,
%U A279767 247,248,265,269,281,299,301,303,306,311,319,321,327,339,340,347,348,391,393,411,413
%N A279767 Numbers m such that m and m+2 have the same prime signature.
%C A279767 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
%H A279767 Amiram Eldar, <a href="/A279767/b279767.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5585 from Michel Marcus)
%H A279767 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%e A279767 18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
%e A279767 19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.
%t A279767 primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)
%o A279767 (PARI) isok(n) = vecsort(factor(n)[,2]) == vecsort(factor(n+2)[,2]); \\ _Michel Marcus_, Feb 25 2018
%Y A279767 Cf. A001359, A052213, A052214.
%K A279767 nonn,easy
%O A279767 1,1
%A A279767 _Altug Alkan_, Dec 18 2016