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Numbers k such that k, k+1, ..., k+5 all have 3 prime factors (counted with multiplicity).
1

%I #19 Aug 17 2024 23:09:24

%S 2522,4921,18241,25553,27290,40313,90834,95513,98282,98705,117002,

%T 120962,136073,136865,148682,153794,181441,181554,185825,211673,

%U 211674,212401,215034,216361,231002,231665,234641,236041,236634,266282,281402,285410,298433,298434,330473,331985,346505,381353

%N Numbers k such that k, k+1, ..., k+5 all have 3 prime factors (counted with multiplicity).

%C First differs from A045942 at position 20, where a(20) = 211673 but A045942(20) = 204323.

%C All terms == 1 or 2 (mod 8).

%C One of the numbers k, k+1, ..., k+5 is a Zumkeller number (A083207), since it is of the form 2*3*p, where p is prime > 3. - _Ivan N. Ianakiev_, Aug 08 2024

%H Robert Israel, <a href="/A375239/b375239.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 18241 is a term because

%e 18241 = 17 * 29 * 37

%e 18242 = 2 * 7 * 1303

%e 18243 = 3^2 * 2027

%e 18244 = 2^2 * 4561

%e 18245 = 5 * 41 * 89

%e 18246 = 2 * 3 * 3041

%e are all products of 3 primes (counted with multiplicity).

%p R:= NULL: count:= 0: p:= 1:

%p while count < 100 do

%p p:= nextprime(p);

%p x:= 4*p;

%p if andmap(t -> numtheory:-bigomega(t)=3, [x-2,x-1,x+1,x+2]) then

%p if numtheory:-bigomega(x-3) = 3 then R:= R, x-3; count:= count+1; fi;

%p if numtheory:-bigomega(x+3) = 3 then R:= R, x-2; count:= count+1; fi;

%p fi;

%p od:

%p R;

%t s = {}; Do[If[{3, 3, 3, 3, 3, 3} == PrimeOmega[Range[k, k + 5]],

%t AppendTo[s, k]], {k, 1000000}]; s

%Y Subsequence of A045942 and of A113789. Contains A259756.

%K nonn

%O 1,1

%A _Zak Seidov_ and _Robert Israel_, Aug 06 2024