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A359746 Numbers k such that k, k+1 and k+2 have the same ordered prime signature.

%I A359746 #22 Jan 17 2023 09:23:50
%S A359746 33,85,93,141,201,213,217,301,393,445,633,697,921,1041,1137,1261,1309,
%T A359746 1345,1401,1641,1761,1837,1885,1893,1941,1981,2013,2101,2181,2217,
%U A359746 2305,2361,2433,2461,2517,2641,2665,2721,2733,3097,3385,3601,3693,3729,3865,3901,3957
%N A359746 Numbers k such that k, k+1 and k+2 have the same ordered prime signature.
%C A359746 First differs from its subsequence A039833 at n = 17, and from its subsequence A075039 at n = 53.
%C A359746 The ordered prime signature of a number n is the list of exponents of the distinct prime factors in the prime factorization of n, in the order of the prime factors (A124010).
%C A359746 Can 4 consecutive integers have the same ordered prime signature? There are no such quadruples below 10^9.
%C A359746 The answer to the question above is no. Two out of every four consecutive numbers are even and their powers of 2 are different. - _Ivan N. Ianakiev_, Jan 13 2023
%H A359746 Amiram Eldar, <a href="/A359746/b359746.txt">Table of n, a(n) for n = 1..10000</a>
%e A359746 33 is a term since 33 = 3^1 * 11^1, 34 = 2^1 * 17^1, and 35 = 5^1 * 7^1 have the same ordered prime signature, (1, 1).
%e A359746 4923 is a term since 4923 = 3^2 * 547^1, 4924 = 2^2 * 1231^1, and 4925 = 5^2 * 197^1 have the same ordered prime signature, (2, 1).
%e A359746 603 is a term of A052214 but not a term of this sequence, since 603 = 3^2 * 67^1, 604 = 2^2 * 151^1, and 605 = 5^1 * 11^2 have different ordered prime signatures, (2, 1) or (1, 2).
%t A359746 q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]
%o A359746 (PARI) lista(nmax) = {my(e1 = [], e2 = factor(2)[,2]); for(n = 3, nmax, e3 = factor(n)[,2]; if(e1 == e2 && e2 == e3, print1(n-2, ", ")); e1 = e2; e2 = e3); }
%Y A359746 Subsequence of A052214 and A359745.
%Y A359746 Subsequences: A039833, A075039.
%Y A359746 Cf. A124010, A175590.
%K A359746 nonn
%O A359746 1,1
%A A359746 _Amiram Eldar_, Jan 13 2023