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A369670 Integers k such that k, k + 1 and 2 k + 1 all have the same number of prime factors, counted with multiplicity.

%I A369670 #9 Mar 17 2025 11:37:23
%S A369670 2,25,34,38,57,93,118,133,145,171,177,201,205,213,218,298,332,334,361,
%T A369670 381,387,394,446,501,507,604,633,694,698,710,805,842,865,878,898,902,
%U A369670 921,1004,1075,1084,1105,1114,1130,1141,1172,1182,1226,1285,1293,1358,1412,1445,1465,1513,1557,1587,1592
%N A369670 Integers k such that k, k + 1 and 2 k + 1 all have the same number of prime factors, counted with multiplicity.
%H A369670 Robert Israel, <a href="/A369670/b369670.txt">Table of n, a(n) for n = 1..10000</a>
%e A369670 a(3) = 34 is a term because 34 = 2 * 17, 34 + 1 = 35 = 5 * 7 and 2 * 34 + 1 = 69 = 3 * 23 all have 2 prime factors.
%p A369670 filter:= proc(n) local t; uses numtheory; t:= bigomega(n); t = bigomega(n+1) and t = bigomega(2*n+1) end proc:
%p A369670 select(filter, [$1..10000]);
%t A369670 s = {}; Do[If[PrimeOmega[k] == PrimeOmega[k + 1] == PrimeOmega[2 k + 1],AppendTo[s, k]], {k, 1000}]; s
%t A369670 Select[Range[2000],Length[Union[PrimeOmega[{#,#+1,2#+1}]]]==1&] (* _Harvey P. Dale_, Mar 17 2025 *)
%Y A369670 Intersection of A045920 and A117360. Contains A188059.
%Y A369670 Cf. A001222.
%K A369670 nonn
%O A369670 1,1
%A A369670 _Zak Seidov_ and _Robert Israel_, Jan 28 2024