cp's OEIS Frontend

A383400 Starting values of maximal runs of at least five integers, each with exactly two distinct prime factors.

%I A383400 #17 May 13 2025 09:59:17
%S A383400 54,91,115,141,158,205,212,295,301,323,391,535,685,721,799,1135,1345,
%T A383400 1465,1535,1711,1941,1981,2101,2215,2302,2425,2641,3865,4411,5461,
%U A383400 6505,6625,6925,7165,7231,7261,7441,7855,7891,8575,9121,9355,9571
%N A383400 Starting values of maximal runs of at least five integers, each with exactly two distinct prime factors.
%C A383400 omega is the function that counts the number of distinct prime factors of a number.
%C A383400 Integers k such that omega(k) = 2 = omega(k+1) = ... = omega(k+4), but omega(k-1) != 2.
%C A383400 Every term is also in A088986, but any subsequent terms from the same run are removed.
%e A383400 a(1) = 54 (run length 5)
%e A383400   53 = 53: omega = 1 (prime, so the previous number is not counted)
%e A383400   54 = 2 * 3^3: omega = 2
%e A383400   55 = 5 * 11: omega = 2
%e A383400   56 = 2^3 * 7: omega = 2
%e A383400   57 = 3 * 19: omega = 2
%e A383400   58 = 2 * 29: omega = 2  (58 is the last member; 59 is prime)
%e A383400 a(2) = 91  (run length 6)
%e A383400   90 = 2 * 3^2 * 5: omega = 3
%e A383400   91 = 7 * 13: omega = 2
%e A383400   92 = 2^2 * 23: omega = 2
%e A383400   93 = 3 * 31: omega = 2
%e A383400   94 = 2 * 47: omega = 2
%e A383400   95 = 5 * 19: omega = 2
%e A383400   96 = 2^5 * 3: omega = 2  (97 is prime, so the run stops at 96)
%e A383400 a(4) = 141  (run length 8)
%e A383400   140 = 2^2 * 5 * 7: omega = 3
%e A383400   141 = 3 * 47: omega = 2
%e A383400   142 = 2 * 71: omega = 2
%e A383400   143 = 11 * 13: omega = 2
%e A383400   144 = 2^4 * 3^2: omega = 2
%e A383400   145 = 5 * 29: omega = 2
%e A383400   146 = 2 * 73: omega = 2
%e A383400   147 = 3 * 7^2: omega = 2
%e A383400   148 = 2^2 * 37: omega = 2  (149 is prime)
%t A383400 With[{s = Select[Range[10000], PrimeNu[#] == 2 &]},
%t A383400   First /@ Select[Split[s, #2 == #1 + 1 &], Length[#] >= 5 &]
%t A383400 ]
%Y A383400 Cf. A001221, A088986, A364307.
%K A383400 nonn
%O A383400 1,1
%A A383400 _IWABUCHI Yu(u)ki_, Apr 25 2025