A359746 - OEIS

%I #22 Jan 17 2023 09:23:50

%S 33,85,93,141,201,213,217,301,393,445,633,697,921,1041,1137,1261,1309,

%T 1345,1401,1641,1761,1837,1885,1893,1941,1981,2013,2101,2181,2217,

%U 2305,2361,2433,2461,2517,2641,2665,2721,2733,3097,3385,3601,3693,3729,3865,3901,3957

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

%C First differs from its subsequence A039833 at n = 17, and from its subsequence A075039 at n = 53.

%C 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 Can 4 consecutive integers have the same ordered prime signature? There are no such quadruples below 10^9.

%C 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 Amiram Eldar, <a href="/A359746/b359746.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 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 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 q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]

%o (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 Subsequence of A052214 and A359745.

%Y Subsequences: A039833, A075039.

%Y Cf. A124010, A175590.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jan 13 2023